DE 


No. 


RATIONAL 
GRAMMAR  SCHOOL  ARITHMETIC 


BY 


GEORGE    W.    MYERS,    Ph.D. 

PROFESSOR  OF  THE  TEACHING   OF  MATHEMATICS  AND  ASTRONOMY,   SCHOOL  OF 
EDUCATION,  THE   UNIVERSITY  OF  CHICAGO 


SARAH    O.   BROOKS 

CITY  NORMAL  SCHOOL,  BALTIMORE,  MD. 


CHICAGO 
SCOTT,    FORESMAN    AND    COMPANY 

1903 


COPYRIGHT,    1903,    BY 
SCOTT,    FORESMAN  AND  COMPANY 

EPUCATION 


tS,  CHICAGO. 


TYPOGRAPHY  BY 
MARSH,  AITKKN  Ji  CURTIS  COMPANY,  CHICAGO 


PREFACE 

The  present  book  is  an  outgrowth  of  the  notion  that  arithmetic 
as  a  science  of  pure  number,  and  arithmetic  as  a  school 
science,  must  be  treated  from  two  essentially  different  stand- 
points. Viewed  as  a  finished  mental  product,  arithmetic  is  an 
abstract  science,  taking  its  bearings  solely  from  the  needs  of  the 
subject;  but  viewed  as  a  school  subject,  arithmetic  should  bean 
abstracted  science,  taking  its  bearings  mainly  from  the  needs  of 
the  learner.  The  former  calls  only  for  logical  treatment,  while 
the  latter  calls  for  psychological  treatment  as  well.  In  other  words, 
to  be  of  high  educational  value  the  school  science  of  arithmetic 
must  take  into  full  account  the  particular  stage  of  the  pupil's 
development.  The  abstract  stage  must  be  approached  by  steps 
which  begin  with  the  learner,  rise  with  his  unfolding  powers,  and 
end  leaving  him  in  possession  of  the  outlines  of  the  science  of 
arithmetic.  To  break  vital  contact  with  the  learner  at  any  stage 
of  the  unfolding  process  is  fatal.  A  controlling  principle  in  the 
development  of  the  various  topics  of  this  book  is  that  any  phase 
of  arithmetical  work,  to  be  of  value,  must  make  an  appeal  to  the 
life  of  the  pupil. 

But  the  social  and  industrial  factors  in  American  communities 
enter  largely  into  the  pupil's  life.  This  renders  material  drawn 
from  industrial  sources  and  from  everyday  affairs  of  high  pedagog- 
ical value  for  arithmetic.  The  recent  infusion  of  new  life  into  the 
curricula  of  elementary  schools  through  the  wide  introduction  into 
them  of  nature  study,  manual  training,  and  geometrical  drawing 
furnishes  a  basis  for  a  closer  unifying  of  the  pupil's  work  in  arith- 
metic with  his  work  in  the  other  school  subjects.  Wide  use  has 
been  made  of  all  these  sources  of  arithmetical  material. 

A  rational  presentation  of  the  processes  and  principles  of  arith- 
metic can  be  secured  as  well  through  material  representing  real 
conditions  as  through  material  representing  artificial  conditions. 
Not  only  have  most  of  the  problems  been  drawn  from  real  sources 


IV  PREFACE 

but  a  yejry  sa-rriest  effort  has  also  been  made  to  have  all  data  of 
'problems  absolutely  correct  and  consistent,  to  the  end  that  infer- 
'frotai  titepo.  may  be  relied  upon.  With  so  rich  a  store  as  the 
corifams,  however,  it  is  perhaps  too  much  to  hope  that  no 
errors  remain.  The  authors  will  deem  it  a  favor  to  be  notified  of 
any  errors  that  may  be  detected. 

It  is  well  known  that  the  majority  of  the  pupils  of  the 
elementary  school  never  reach  the  high  school.  Even  these 
pupils,  whose  circumstances  cut  them  off  from  advanced  mathe- 
matical study,  have  a  right  to  claim  of  the  elementary  school  some 
useful  knowledge  of  the  more  powerful  instruments  of  algebra 
and  geometry.  For  those  who  will  continue  their  studies  into 
the  high  school  it  is  important  that  the  roots  of  the  later  mathe- 
matical subjects  be  well  covered  in  the  soil  of  the  earlier.  The 
present  book  seeks  to  meet  the  needs  of  both  classes  of  pupils 
through  the  organic  correlation  of  the  elements  of  geometry  and 
algebra  with  the  arithmetic  proper.  Treated  thus,  the  geometry 
serves  to  illustrate  the  work  of  arithmetic  and  algebra,  and  the 
algebra  emerges  from  the  arithmetic  as  generalized  number. 

In  particular,  this  text  aims  to  accomplish  four  main  pur- 
poses, viz. : 

(1)  To  present  a  pedagogical  development  of  elementary  mathe- 
matics, both  as  a  tool  for  use  and  as  an  elementary  science ; 

(2)  To  base  this  development  on  subject-matter  representing 
real  conditions; 

(3)  To  open  to  the  pupil  a  wide  range  and  variety  of  uses  for 
elementary  mathematics  in  common  affairs — to  aid  him  to  get  a 
working  hold  of  his  number  sense ;  and 

(4)  To  give  the  pupil  some  training  in  ways  of  attacking  com- 
mon problems  arithmetically  and  some  power  to  analyze  quanti- 
tative problems. 

It  is  not  intended  that  teachers  should  have  their  classes  solve 
all  the  problems;  but  rather  that  each  teacher  should  select  such 
topics  and  problems  as  have  a  particular  interest  for  his  school. 
We  trust  that  the  suggestiveness  of  this  book  as  to  sources  of 
problems  and  ways  of  handling  them  in  the  arithmetic  class  will 
be  appreciated  by  many  teachers.  Such  teachers  will  doubtless 


PREFACE  T 

prefer  to  work  out  some  topics  more  fully  than  is  done  in  the  text. 
It  is  thought  that  what  is  given  in  the  text  will  make  the  fuller 
working  out  of  special  topics  practicable  and  not  difficult. 

The  work  makes  continual  call  for  estimating  magnitudes  and 
for  actual  measurement  by  the  pupils.  Let  the  children  be  sup- 
plied with  measures,  foot-rules,  yardsticks,  meter-sticks,  etc.,  and 
encourage  their  constant  use.  Make  so  regular  a  feature  of  this 
work  that  pupils  form  the  habit  of  estimating  distances,  areas, 
volumes,  weights,  etc.,  always  correcting  their  estimates  by  actual 
measurement. 

All  models,  scales,  or  standards  of  measure  made  by  pupils 
should  be  carefully  kept  and  used  in  the  later  work. 

Pupils  should  also  be  supplied  with  instruments  with  which 
the  exercises  in  constructive  geometry  may  be  actually  done.  A 
cheap  pair  of  compasses  is  the  only  necessary  purchase.  It  is 
better  that  the  pupil  make  the  rest  of  the  apparatus  needed. 
If  for  any  reason  the  pupil  cannot  actually  do  the  constructive 
work  it  should  be  omitted  altogether. 

The  Introduction  which  precedes  the  work  in  the  formal 
operations  is  a  departure  from  current  text-book  procedure  believed 
to  be  worthy  of  special  remark.  It  consists  of  thirty  pages  of 
simple,  practical  problems,  relating  to  matters  with  which  the 
pupil's  experiences,  in  school  and  out  of  school,  have  familiarized 
him  in  an  indefinite  way,  for  the  right  understanding  of  which  the 
use  of  numbers  and  of  the  arithmetical  processes  is  necessary. 
This  chapter,  by  furnishing  to  the  pupil  a  gradual  transition  from 
his  vacation  experiences  to  the  rather  severe  study  of  the  formal 
processes  of  arithmetic,  by  impressing  the  pupil  with  a  sense  of 
the  real  need  and  purpose  of  such  study,  thus  preparing  his  inter- 
est for  it,  will  be  recognized  as  a  deviation  from  common  practice 
resting  upon  sound  pedagogical  grounds. 

Especial  attention  is  invited  to  this  introductory  chapter;  to 
the  chapter  on  measurement,  §§76-86,  preceding  and  laying  the 
foundation  for  common  fractions;  to  the  treatment  of  propor- 
tion, §§90-92,  125,  126 ;  to  the  lists  of  data  for  original  problems, 
§§36,  42,  50,  140;  to  the  work  in  geography,  §§30,  34,  60,  166; 
in  farm  account  keeping,  §§17,  41;  in  commerce,  §35;  in  nature 


VI  PREFACE 

study,  §§51,  101,  130,  132,  164,  165,  190,  191;  in  physical 
measurements,  §§8-10,  133,  141;  in  paper-folding,  §183;  on 
the  locomotive  engine,  §175;  in  constructive  geometry,  §§75, 
177-188,  197;  and  in  applied  algebra,  §217.  Most  teachers 
will  approve  dropping  many  of  the  time-honored  but  anti- 
quated subjects  and  the  curtailing  of  other  topics  of  slight 
utility,  and  the  introduction  in  their  places  of  more  timely  and 
more  real  subjects.  The  many  lists  of  data  for  original  problems, 
§§36,  42,  50,  140;  the  treatment  of  longitude  and  time,  §§192, 
193,  the  practical  work  leading  up  to  this  topic;  the  extensive  use 
of  graphs  to  put  meaning  into  arithmetical  measures  and  to 
bring  out  the  laws  involved  in  numerical  data  will  commend 
themselves  to  teachers  as  useful  and  instructive  means  of  keep- 
ing the  number  faculty  employed  on  practical  material. 

Another  special  feature  is  the  numerous  lists  of  problems  that 
bear  on  the  development  of  some  important  idea  or  law,  having 
an  interest  on  its  own  account.  In  these  lists  each  problem  is  a 
step  in  a  connected  line  of  thought  culminating  in  an  important 
truth.  This  plan  furnishes  numerous  problems,  miscellaneous  as 
to  process,  thereby  requiring  original  mathematical  thought,  and 
still  organically  related  to  a  central  idea,  thereby  calling  for  the 
constant  exercise  of  judgment.  Examples  of  this  may  be  seen  in 
any  section  of  the  introduction;  in  the  problems  on  geography, 
commerce,  nature  study,  and  elementary  science  and  also  in  the 
following  sections:  72,  73,  76-89,  101,  118,  123-126,  141,  144, 
145,  164,  165,  166,  168,  169,  and  in  practically  all  the  matter 
from  p.  271  to  the  end  of  the  book.  For  the  maturity  of  pupils 
of  the  later  grades  this  is  believed  to  be  an  important  feature. 
It  avoids  the  danger,  always  present  with  lists  of  promiscuous 
problems  when  classified  under  the  arithmetical  processes  to  be 
exemplified,  of  reducing  to  the  mechanical  what  should  never  be 
allowed  to  become  mechanical,  viz. :  the  analysis  of  relations. 
Throughout  the  book  the  instruction  is  addressed  to  the  pupil's 
understanding  rather  than  to  his  memory. 

But  while  accomplishing  this,  due  regard  has  been  had  to  the 
necessity  of  sufficient  drill  in  pure  number  to  enable  the  pupil  to 
obtain  both  a  conscious  recognition  of  processes  and  considerable 


PREFACE  '».*::  ^2    V         vil 


facility  in  their  automatic  use.  This  is  done&ut 
that  the  fundamental  arithmetical  operations  should  be  reduced 
to  the  automatic  stage  as  early  as  possible,  consistently  with  a 
clear  understanding  of  them. 

The  attention  of  teachers  is  called  to  the  section  on  Short 
Methods  and  Checking  at  the  close  of  the  book.  After  pupils 
have  clearly  grasped  the  meaning  of  the  arithmetical  processes  and 
have  acquired  some  mastery  of  their  uses,  a  relief  from  the  tedium 
of  long  arithmetical  calculations  becomes  a  matter  of  great  impor- 
tance. Xo  one  can  become  a  rapid  computer  without  short 
methods  and,  considering  that  the  problems  of  daily  life  that  call 
for  arithmetical  treatment  do  not  have  answers  with  them,  no  one 
can  be  certain  of  his  results  without  means  of  checking  calcula- 
tions. Every  expert  accountant  uses  them  and  the  more  expert 
he  is  the  more  does  he  use  them.  In  fact,  expertness  consists  very 
largely  in  the  ability  to  shorten  calculations  and  to  apply  rapid 
checks.  Training  in  the  use  of  short  cuts  and  checks  should  con- 
stitute a  much  more  important  part  of  the  pupil's  work  in  arith- 
metic than  is  common.  Constant  use  of  these  sections  should  be 
made  through  the  seventh  and  eighth  grades. 

Definitions,  processes,  rules,  and  even  special  subjects  are 
worked  out  under  the  guidance  of  the  principle:  "First  its 
informal,  though  rational,  use,  and  afterwards  conscious  recogni- 
tion of  its  formal  use."  Definitions  of  merely  technical  terms  are 
given  when  called  for  by  the  development  and  where  the  need  for 
them  arises. 

Unusual  attention  has  been  paid  to  the  numerous  illustrations. 
The  publishers  have  spared  no  pains  to  make  them  an  important 
aid  to  the  teacher  in  the  development  of  the  subject.  None  have 
been  inserted  for  mere  adornment  and  none  are  mere  pictures  of 
things  or  relations  that  are  perfectly  obvious  to  the  pupil.  Their 
aim  throughout  is  to  secure  clearness,  precision,  and  certainty  of 
thought.  It  is  thought  the  book  may  lay  rightful  claim  to  an 
innovation  in  this  particular. 

For  convenience  in  reviews  and  for  reference,  a  synopsis  o'f 
definitions  and  a  full  index  are  given. 

The  authors'  acknowledgments  for  suggestions  and  ideas  are 


PREP  ACE 


ibxuumetotts  arithmetical  writers  and  teachers.  They  desire 
to  express  their  obligations  in  particular  to  Miss  Katheriiie  M. 
Stilwell  of  the  School  of  Education,  University  of  Chicago,  for 
valuable  help  throughout  the  book;  to  Mr.  W.  S.  Jackman, 
Dean  of  the  School  of  Education  (to  whom  is  due  the  credit  for 
the  skiameter  work  as  given  in  §190,  p.  299),  for  the  privilege 
of  using  much  valuable  material;  to  Mr.  J.  B.  Eussell,  Super- 
intendent of  Schools,  Wheaton,  111.,  and  to  Miss  Ada  Van  Stone 
Harris,  Supervisor  of  Primary  Schools,  Rochester,  N.  Y.,  both  of 
whom  read  much  of  the  proof.  They  desire  most  heartily  to 
thank  Mr.  Stephen  Emery  of  Lewis  Institute,  whose  unceasing 
diligence  and  pains  with  the  proofs  have  wrought  distinct  improve- 
ments on  nearly  every  page  of  the  book. 

If  the  book  shall  in  some  measure  aid  in  putting  the  teaching 
of  arithmetic  on  a  more  rati  nal  basis,  thereby  bringing  a-bout 
results  more  nearly  c  mmensurate  with  the  time  and  energy  put 
upon  the  subject  in  the  elementary  schools,  the  authors  will  deem 
their  efforts  repaid.  THE  AUTHORS. 

CHICAGO,  July,  1903. 


RATIONAL 
GRAMMAR  SCHOOL  ARITHMETIC 


INTRODUCTION 


ORAL    WORK 


FIGURE  l 


Bookcase 


§1.  Scale  Drawing. 

1.  If  one  side  of  a  triangle  is  4  in.  long,  how  long  a  line  will 
represent   this  side   in  a   drawing   to  a   scale   of   1    to  4,  or  £? 

2.  If  the  two  other  sides  are  2  in.  and  3 
in.  long,  how  long  must  the  lines  be  to  rep- 
resent  them    in   the   same  drawing?    in   a 
drawing  to  a  scale  of  -j^? 

3.  What   is  the  scale   of  a   drawing   in 

which  a  line  1  in.  long  represents  1  ft.?  in  which  1  in.  represents 
40  ft.?  100  ft.? 

4.  In  Fig.  2,  which  is  a  scale 
drawing   of  a  schoolroom,  one- 
sixteenth  of  an  inch  represents 
1  ft.     How  long  is  the   room? 
the  teacher's  desk?     How  long 
and   how  wide   are   the  pupils' 
desks? 

5.  In  what  direction  do  the 
pupils  face,  when  seated  at  their 
desks? 

6.  Find  by  measurement  how 
far   it   is    from    the  northwest 

corner  of  desk  1  to  the  southwest  corner  of  desk  4;   from  the 
west  edge  of  desk  2  to  the  east  wall  of  the  room. 

7.  Make  and  answer  other  questions  on  this  plan. 

8.  Make  a  similar  scale  drawing  from  actual  measurements  of 
your  schoolroom  and  the  fixed  objects  within  it. 

9.  Make  a  scale  drawing  from  measurements  of  your  school- 
house  and  grounds. 

.1 


i 


G 


FIGURE  2 


-.2: 


&AXIONAL   GRAMMAR    SCHOOL   ARITHMETIC 


;§£  ;T(*wii  Blofck Cartel  Lots. — Fig.  3  represents  a  scale  map  of  a  town 
block.  Scale:  1  in.  equals  100  ft.  If  possible,  pupils  should 
measure  and  draw  to  scale  a  block  or  field  in  the  neighborhood 
of  the  schoolhouse,  and  in  all  problems  use  the  numbers  they 
obtain  from  their  own  measurements  in  preference  to  those  given 
in  the  exercises. 


RACE 


ST. 


B 


K 


G 


H 


w- 


MARXET 


ST. 


.1  in.  divided  into  tenths 


FIGURE  3 


Measure  the  drawing  in  Fig.  3,  and  find  how  long  the  block 
is  between  the  sidewalks;  how  wide. 

NOTE. — Use  the  scale  of  tenths  given  in  the  figure.  Lay  a  strip  of 
paper  having  a  straight  edge  beside  the  marked  inch  and  mark  short 
lines  on  the  strip  to  indicate  the  tenths. 


INTRODUCTION  3 

WRITTEN    WORK 

1.  How  many  square  feet  in  the  area  of  the  block  mnop? 

2.  At  $20  per  ft.  of  frontage  on  Market  street,  what  is  lot  L 
worth? 

3.  Make  problems  like  2  for  other  lots  on  Market  street,  using 
price  per  front  foot  of  lots  where  you  live. 

4.  At  $12.50  per  ft.  of  frontage  on  Mathews  avenue,  what  is 
lot  H  worth? 

5.  Make  similar  problems  for  other  lots  on  Mathews  avenue. 

6.  At  $18  per  ft.  of  frontage  011  Race  street,  what  is  lot  G 
worth?     Similar  problems  should  be  made  by  the  pupil. 

7.  Make  problems  for  any,  or  all,  of  the  lots,  at  the  price  per 
front  foot  where  you  live. 

Each  property  holder  is  taxed  to  provide  funds  for  founda- 
tion material,  brick,  and  labor  to  pave  in  front  of  his  property  to 
the  middle  line  of  the  street.  This  is  called  an  assessment. 

8.  The  streets  are  to  be  paved  with  brick.     The  cost  of  exca- 
vating to  the  proper  depth  is  30^-  per  square  yard  of  surface;*  find 
the  cost  of  excavating  a  strip  1  yd.  wide  extending  from  outside 
edge  of  sidewalk  to  middle  of  Race  street.  Ans.  $3.00. 

9.  What  would  be  the  cost  of  excavating  a  similar  strip  5  yd. 
wide?  a  strip  as  wide  as  frontage  of  lot  E?     Ans.  $15.00 ;  $50.00. 

10.  The  cost  of  foundation  material  is  36^  per  square  yard ; 
find  the  cost  of  enough  such  material  for  the  strip  described  in 
problem  8.  Ans.  $3.60. 

11.  Eind  the  cost  of  foundation  material  for  each  of  the  two 
strips  mentioned  in  problem  9.  1st  Ans.  $18.00. 

12.  The  cost  of  the  brick  to  oe  used  is  $10  per  M  (thousand) 
and  78  bricks  are  needed  to  cover  1  sq.  yd.     What  is  the  cost  of 
the  brick  needed  for  the  strip  of  problem  8?   for   each  of   the 
two  strips  of  problem  9?  1st  Ans.  $7.80. 

13.  What  is  the  total  cost  of  excavating,  of  foundation  mate- 
rial, and  of  brick  for  the  strip  of  problem  8?  for  each  of  the  two 
strips  of  problem  9?  1st  Ans.  $14.40. 

*  Use  prices  current  in  your  community  whenever  they  can  be  obtained,  instead  of 
prices  given. 


4  RATIONAL    GRAMMAR    SCHOOL    ARITHMETIC 

14.  Find  this  total  for  other  lots  fronting  on  either  Market 
or  Eace  street. 

15.  The  labor  of  construction  costs  75^  per  sq.  yd.     What  is 
the  cost  of  labor  on  a  strip  of  the  size  mentioned  in  problem  8? 

Am.  $7.50. 

16.  What  will  be  the  assessment  against  lot  F  for  paving  for 
each  foot  of  frontage? 

17.  Make  similar  problems  for  other  lots,  not  including  the 
corner  lots. 

18.  Henry    and    Mathews    avenues,    which    are     30  ft.  wide 
between  sidewalks,  are  to  be  paved  in  the  same  way  as  Market 
and  Race  streets.     Rates  being  the  same  as  above,  what  will  be  the 
total  assessment  against  lot  L? 

19.  Make  and  solve  problems  like  18  for  other  lots. 

20.  The  owner  builds  a  house  on  lot  L.     The  length  of  the 
house  is  36  ft.  and  the  width  is  30  ft.     How  many  square  feet  of 
the  lot  are  covered  by  the  house? 

21.  The  southeast  corner  of  the  house  is  located  30  ft.  from 
the  south  and    east  lines  of  the   lot.     Locate   the   three   other 
corners. 

22.  Find  the  cost,  at  45^  per  sq.  ft.,  of  the  concrete  walk  3 
ft.  wide  in  lot  L,  as  shown  in  the  drawing. 

§3.  House  and  Furnishings.     ORAL  WORK 

1.  What  is  meant  by  a  1-brick  wall?*  a  2-brick  wall? 

2.  If  it  takes  7  bricks  to  make  1  sq.  ft.  of  wall  surface  when 
laid  in  a  1-brick  wall,  how  many  bricks  per  square  foot  are  needed 

for  a  2-brick  wall?   a  3-brick  wall?  a  5- 
brick  wall? 

3.  From  the  dotted  lines  in  Fig.  4, 
can  you  tell  how  measurements  may  be 
taken  on  a  wall  to  avoid  counting  corners 
twice? 


OHtside-Measure 

pwriTT},,  4  4.  A  brick  is  2  in.  by  4  in.  by  8  in. ; 

•cTGlTH/E  4  J  J 

how  many  cubic  inches  are  there  in  it? 

*  A  1-brick  wall  is  a  wall  one  brick  thick,  bricks  lying  on  the  largest  surfaces. 


INTRODUCTION 


5 


5.  How  many  square  inches  in  one  end?   one  edge?  one  of  its 
largest  surfaces? 

6.  In  locating  the  foundation  of  the  house  on  lot  L  (Fig.  3), 
in  what  direction  would  you  wish  the  line  of  the  front  foundation 
wall  to  run  with  reference  to  the  Market  street  line? 

7.  Point  out    some   of    the  square  corners    on  the   plans    in 
Fig.  5. 

8.  Can  you  point  out  any  corners  that  are  not  square? 


WRITTEN    WORK 

The  owner  of  lot  L,  which  fronts  on  Market  street,  builds  the 
house  whose  foundation  plans  are  given  in  Fig.  5  and  whose  first 
and  second  floor  plans  are  given  in  Fig.  6. 


FOUNDATION 


FlGFRB  5 


WALLS 


1.  What  will  it  cost  to  excavate  a  rectangle  21'  x  36'  to  a  depth 
of  6  ft. ,  for  the  foundations  and  cellar,  at  20^  per  cubic  yard? 

Think  of  the  foundation  as  made  up  of  straight  walls  such  as 
are  shown  in  the  second  part  of  Fig.  5.  The  mark  (')  means  foot 
or  feet,  and  (")  means  inch  or  inches. 

2.  The   foundation  is   inclosed  by  2  side  walls,  each   36  ft. 
long,  and  2  end  walls,  each  18J  ft.  long.     Point  out  these  walls 
in  both  parts  of  Fig.  5.     The  foundation  walls  are  all  8  ft.  high. 
Find  the  area  of  the  north  surface  of  the  north  side  wall. 

Ans.  288  sq.  ft. 

3.  What  other  wall  has  an  outer  surface  equal  to  the  surface 
mentioned  in  problem  2? 


6  RATIONAL    GRAMMAR,   SCHOOL    ARITHMETIC 

4.  Find  the  area  of  the  outer  surface  of  the  east  end  wall. 

Ans.  148  sq.  ft. 

5.  What  other  outside  surface  equals  this  one  in  area? 

G.  Masons  reckon  that  it  takes  14  bricks  for  each  square  foot 
of  outer  surface  to  lay  a  solid  2-brick  wall.  If  all  walls  are 
2-brick  walls,  and  solid,  how  many  bricks  will  be  needed  for  the 
north  foundation  wall?  for  the  east  wall?  the  south?  the  west? 

1st  Ans.  4032;  2d  Ans.  2072. 

7.  How  many  bricks  will  be  needed  for  all  four  of  the  outside 
walls? 

8.  There  are  two  inside  foundation  walls  each  18£  ft.  long,  also 
one  10^  ffc.  long,  and  one  11  ft.  long.     All  are  2-brick  walls  8  ft. 
high.     If  these  walls  contain  no  openings,  how  many  bricks  will  be 
needed  for  the  inside  foundation  walls?  Ans.  6552. 

9.  The  outside  walls  contain  8  window  openings  each  1^-  ft.  by 
3  ft.     If  masons  allow  for  one-half  the  area  of  all  openings  in 
computing  the   number  of  bricks,  how  many  bricks  should  be 
deducted  for  the  outside  walls?  Ans.  252. 

10.  The  inside  walls  contain  4  door  openings  each  2^-  ft.  by  8 
ft.     How  many  bricks  should  be  deducted  for  the  inside  walls? 

Ans.  560. 

11.  Find  the  total  number  of  bricks  needed  for  the  foundation 
walls  and  their  cost  at  $9  per  M.* 

12.  If  hauling  costs  75^  per  load  of  1^  T.  and  each  brick  weighs 
6  lb.,  find  the  cost  of  hauling  the  brick  for  the  foundations.! 

13.  If  the  mortar  costs  $1.25  per  M.   bricks,  what  will  be  the 
cost  of  the  mortar? 

14.  Four  brick  piers  2  ft.  by  2  ft.  and  4  ft.  high  support  the 
porch  columns.     How  much  will  the  bricks  needed  for  these  col- 
umns cost  at  $12  per  M.,  counting  22-J  bricks  for  each  cubic 
foot? 

15.  While  the  house  was  building  the  owner  decided  to  replace 
weather -boarding  by  stained  shingles  on  a  belt  running  around 
the  house  and  extending  to  a  distance  of  9  ft.  below  the  eaves. 
Each  square  yard,  thus  changed,  cost  $1.75  extra,  no  allowance 

*  In  computing  the  cost  of  brick  use  the  nearest  whole  thousand, 
t  When  the  last  load  is  fractional,  the  price  for  a  full  load  is  charged. 


INTRODUCTION  7 

being  made  for  openings.     How  much  does  this  change  add  to  the 
cost  of  the  house? 

16.  Hardwood  floors  were  decided  upon  later  to  take  the  place 
of  pine  floors  in  the  dining-room  and  front  hall.  This  increased 
the  price  by  !%<p  per  sq.  ft.  How  much  did  this  add  to  the  cost  of 
the  house,  counting  the  dining-room  14  ft.  G  in.  wide? 

Am.  $51.54. 


Il'6"xt6'6"l  BEDROOM  DRESS- 


FIRST   FLOOR 


SECOND  FLOOR 


FIGURE  6 


17.  A  room  is  said  to  be  well  lighted  when  the  floor  area  is  not 
more  than  G  times  the  area  of  the  window  surface  which  admits 
light.     If  all  windows  as  shown  in  the  plan  are  2J  ft.  wide  and  G 
ft.  high,  and  \  of  the  window  space  is  covered  by  the  sash,  is  the 
dining-room  well  lighted?  is  the  kitchen?  the  hall? 

18.  Is  your  schoolroom  well  lighted?     Are  the  halls  of  your 
schoolhouse  well  lighted? 

§4.  Cost  of  Living.  ORAL  WORK 

1.  If  the  living  expenses  of  a  family  are  $70  per  mo.,  what  is 
the  expense  per  year? 

2.  If  house  rent  is  $25  per  mo.  and  other  expenses  are  $50, 
what  part  of  the  total  expense  is  house  rent? 

3.  How  many  work  days  are  there  in  a  month?     How  many 
weeks  in  a  year? 

4.  If  a  man's  car  fare  amounts  to  20^  a  d.  for  26  d.  a  mo., 
what   is  his  monthly  expense  for  car  fare?     What  is  his  yearly 
expense? 


8 


RATIONAL    GRAMMAR    SCHOOL    ARITHMETIC 


WRITTEN   WORK 

1.  Find  out  at  home  or  from  your  neighbor  how  much  hard 
coal  is  required  to  supply  some  furnace  or  stove  1  mo.     Supposing 
hard  coal  costs  $7.50  per  T.,  how  much  will  the  coal  cost  for  the  6 
winter  months  from  November  to  May? 

2.  If  soft  coal  costs  $3.75  a  T.,  and  If  T.  goes  about  as  far  as 
1  T.  of  hard  coal,  about  how  much  would  it  cost  to  supply  this  fur- 
nace with  soft  coal  for  the  6  winter  months? 

The  prices  given  below  are  fair  averages  for  a  large  city.  They 
would  be  less  in  smaller  places,  and  prices  current  in  your  com- 
munity are  to  be  preferred  to  these. 


GROCER'S  PRICES 

Apples pk.  $.35 

Apricots can  .  25 

Baking  powder. . .  .\  Ib.  can  .20 

Bread loaf  .05 

Butter Ib.  .28 

Cabbage Ib.  2c.,  head  JO 

Cauliflower head  .15 

Celery bunch  .  10 

Cheese Ib.  .15 

Coffee Ib.  .30 

Crackers Ib.  .10 

Eggs doz.  .  25 

Flour 25  Ib.  sack  .50 

Lemons doz.  .25 

Lettuce bunch  .05 

Lima  beans can  .  10 

Milk qt.  .06i 

Oatmeal 2  Ib.  pkg.  .10 

Olives :..pt.  .30 

Onions pk.  .20 

Oranges doz.  .30 

Peaches can  .25 

Pears can  .  25 

Peas can  .12| 

Pickles doz.  .10 

Potatoes pk.  .25 

Prunes Ib.  .10 

Rice Ib.  .081 

NOTE. — It  will  be  assumed  that 
small  quantities. 


GROCER'S  PRICES— Continued 

Rolls doz.  $.10 

Salt Ib.  .05 

Soap bar  .  05 

Starch Ib.  .10 

Sugar Ib.  .05 

Tea Ib.  .60 

Tomatoes can  .  15 

BUTCHER'S  PRICES 

Bacon Ib.  $.20 

Chicken Ib.  .14 

Fish,  fresh Ib.  .12$ 

Fish,  salt Ib.  .10 

Ham Ib.  .18 

Lard Ib.  .13 

Lobsters,  shrimps,  etc. .  .Ib.  .25 

Mutton Ib.  .14 

Oysters qt.  .30 

Pork Ib.  .10 

Pork  tenderloin Ib.  .20 

Porterhouse  steak Ib.  .18 

Round  steak Ib.  .10 

Salmon Ib.  .20 

Sausage  meat Ib.  .10 

Sirloin  steak Ib.  .16 

Spare  ribs Ib.  .08^ 

Turkey,  duck,  etc Ib.  .12£ 

Veal Ib.  .14 

rates  are  the  same  for  large  and 


INTRODUCTION 


Make  such  problems  as : 

3.  If  you  purchase  of  a  grocer  2  loaves  of  bread,  1  Ib.  of  butter, 
1  Ib.  of  coffee,  and  1  pk.  of  potatoes,  and  give  him  $1,  how  much 
change  should  you  receive,  if  prices  are  as  quoted  in  the  table 
above? 

4.  If  you  buy  a  2-lb.  sirloin  steak  and  give  the  butcher  50^, 
what  change  should  you  receive,  prices  being  as  in  the  table? 

5.  Pupils  may  make  out  lists  in  the  form  of  a  statement  arid 
find  the  change  due  if  the  account  is  paid  with  $2,  $5,  or  $10, 
thus : 


BOUGHT 

PAID 

Item 

Item 

1.     2  doz.  rolls          @  lOc. 

$0.20 

1.     Cash                                  $5.00 

2.     1  doz.  eggs                25c. 

.25 

3.     1  doz.  oranges          30c. 

.30 

4.     2  bunches  celery      lOc. 

.20 

5.     1  Ib.  tea                     60c. 

.60 

6.     2  cans  peas              12£c. 

.25 

7.     1  Ib.  starch                lOc. 

.10 

8.     5  bars  soap                  5c. 

.25 

9.     1  pkg.  oatmeal         lOc. 

.10 

Total  cost 

Balance  due  in  change 

6.  Foot  the  statement  above. 

7.  Make  out  a  bill  of  supplies  such  as  a  hotel  keeper  might 
purchase  at  the  grocer's  or  the  butcher's. 

8.  Pupils  may  make  and  solve  problems  similar  to  the  above, 
using  local  prices  when  convenient. 

9.  The  actual  living  expense  account 
of  a  family  for  January  is  as  shown 
in  the  table.      If  this  is  a  fair  average 
monthly  expense  account  for  the  fam- 
ily, what  will  be  its  living  expense  for 
one  year? 

10.  The  family  pays  $3000    for   a 
home   and    is   thereby  saved    $25  per 


Jan.,  1902. 

Meat  

$18.50 

Groceries  . 

27.75 

Milk  

1.88 

Butter  .... 

2.60 

Soft  coal  .  . 

9.65 

Clothing  .  . 

15.00 

10 


RATIONAL    GRAMMAR    SCHOOL   ARITHMETIC 


mo.  in  rent.     The  yearly  rent  amounts  to  what  part  of  the  cost 
of  the  home? 

11.  If  the  home  is  in  the  suburbs  of  a  city  and  the  man  must 
pay  100  morning  and  evening  6  d.  per  wk.  for  car  fare,  how  much 
does  this  add  to  his  yearly  expense  account? 

The  table  given  here  shows  the  average  yearly  cost  to  one 
person  for  food,  clothing,  and  household  utensils  from  1897  to 
1901  inclusive: 


FOB  YEAR 

ENDING 

Bread- 
stuffs 

Meat 

Dairy  & 
Garden 

Other 
Food 

Clothing 

Metals 

Miscel- 
laneous 

Totals 

Dec.  31,  1897 
Dec.31,  1898 
Dec.31,  1899 
Dec.31,  1900 
Dec.31,  1901 

§13.51 
13.82 
13.25 
14.49 
20.00 

17.34 
7.52 
7.25 
8.41 
9.67 

$12.37 
11.46 
13.70 
15.56 
15.25 

18.31 
9.07 
9.20 
9.50 
8.95 

§14.65 
14.15 
17.48 
16.02 
15.55 

$11.57 
11.84 
18.09 
15.81 
1.38 

$12.11 
12.54 
16.31 
15.88 
16.79 

| 

Sums.    . 

Averages  .  .  . 

NOTE. — In  the  above  table  breadstuff s  include  wheat,  corn,  oats,  rye, 
barley,  beans,  and  peas;  meat  includes  lard  and  tallow;  dairy  and  garden 
products  include  vegetables,  milk,  butter,  eggs,  and  fruit;  the  miscel- 
laneous articles  include  a  variety  of  things  which  make  a  part  of  the 
cost  of  living  for  the  average  family. 

12.  Fill  out  the  "totals"  column  on  the  right,  thus  finding  the 
total  cost  of  living  for  1  person  for  each  of  the  5  years. 

13.  What  was  the  increase  in  cost  of  living  from  1897  to  1901? 

14.  Has  the  cost  of  any  of  the  items  decreased  during  this 
period?     How  much  has  been  the  change  in  cost  in  each  case? 

15.  What  fractional  part  of  the  total  cost  of  living  for  the  year 
is  the  cost  of  breadstuffs  in  1897?  in  1901? 

16.  Make  other  problems  like  15. 

NOTE. — The  average  of  any  five  numbers  is  £  of  their  sum. 

17.  Find  the  average  yearly  cost  of  living  for  the  5  years. 

18.  Find  the  average  cost  of  each  item. 

19.  Fill  out  a  column  of  totals  with  the  sums  of  the  numbers 
in  the  second,  third,-  fourth  and  fifth  columns.     What  does  each 
of  the  5  totals  mean? 

20.  Give  the  change,  from  year  to  year,  in  the  total  cost  of 
foods  for  one  person  by  finding  the  difference  between  each  total 
and  the  one  next  after  it. 


INTRODUCTION 


11 


Corn 

Wheat 

Oats 

W-i-E 

III 
S 

e> 

House 

^ 

Oats 

and          /^%^ 

Grounds 

'///!&'  \iM 

§5.  Fencing  a  Farm. — A  farm  was  divided  into  fields  and  seeded 
as  shown  in  the  drawing  (Fig.  7).  The  scale  of  the  drawing  is  1 
in.  to  80  rd.  This  means  that  1  in.  in  the  drawing  represents 
80  rd.  in  the  farm,  and  that  all  other  lines  longer  or  shorter  than 
1  in.  represent  distances  in 
the  farm  proportionately 
longer  or  shorter  than  80  rd. 

ORAL  WORK 

1.  How  long  is  the  corn- 
field north  and  south?  how 
wide? 

2.  How   long    and   how 
wide  is  the  wheatfield?  the 
north  oatfield? 

3.  How   long   and   how 
wide   is   the   meadow?    the 
south  oatfield? 

4.  The  drawing  of   the 
road   is   TV  in.   wide;    how 
wide  is  the  road? 

5.  How  long  and  how  wide  is  the  farm? 

6.  How  long  must  a  wire  be  to  reach  entirely  round  the  farm? 

7.  How  many  rods  of  barbed  wire  will  be  needed  to  inclose  the 
farm  with  a  3-wire  fence?  how  many  miles  (320  rd.  =  1  mile)? 

WRITTEN    WORK 

1.  How  many  rods  long  is  the  partition  fence  running  across 
the  farm  from  east  to  west?  what  part  of  a  mile  is  its  length?    How 
many  rods  of  wire  will  make  it  a  4-wire  fence?        1st  Ans.  160. 

2.  How  many  rods  of  wire  will  run  a  4-wire  fence  across  the 
farm  from  the  middle  of  the  north  side  to  the  middle  of  the  south 
side?  Ans.  640. 

3.  How  many  rods  of  wire  will  run  a  4-wire  fence  between  the 
wheatfield  and  the  north  oatfield?   between  the  meadow  and  the 
south  oatfield? 

4.  How  many  bales  of  wire  of  100  rd.  each  will  be  needed  for 
these  cross  fences  and  division  fences?  Ans.  19.2. 


FIGURE  7 


12  RATIONAL    GRAMMAR    SCHOOL    ARITHMETIC 

5.  How  much  will  the  wire  cost  for  the  3-wire  inclosing  fence 
@  $3.50  a  bale  of  100  rods?  Ans.  $67.20. 

6.  The  posts  for  all  the  barbed  wire  fences  are  set  1  rd.  apart. 
How  many  posts  will  be  needed  to  fence  around  the  farm? 

7.  How  much  will  these  posts  cost  @  25^  apiece? 

8.  How  much^  will  the  wire  cost  @  $2.60  a  bale  for  the  two 
4-wire  cross  fences,  and  the  two  division  fences  between  the  wheat 
and  the  north  oatfield  and  between  the  meadow  and  south  oat- 
field? 

9.  How  much  will  the  posts  for  these  fences  cost  @  15^  apiece? 

10.  Find  the  total  cost  of  wire  and  posts  for  these  fences. 

11.  The  house  and  barn  lot  is  separated  from  the  pasture  by  a 

5-board  fence.  The  boards  used  are 
1  in.  thick,  6  in.  wide,  and  12  ft. 
long.  How  many  boards  will  reach 
once  along  the  north  side  of  the 
house  and  barn  lot?  Ans.  55. 

TWO  FENCE  PANELS  ^  H()W  m      ^oax^  rf^   reaCQ 

r  IGURE  o  * 

once  along  the  east  side  of  the  lot? 

13.  How  many  boards  will  be  needed  for  the  5-board  fence 
along  both  sides? 

14.  The  posts  are  set  6  ft.  between  centers,  that  is,  a  post  is 
set  every  6  ft.  (Fig.  8) .     How  many  posts  will  be  needed  for  the 
north  side?  for  the  east  side?  for  both? 

1st  Ans.  110;  2d  Ans.  109. 

15.  How  much  will  these  posts  cost  @  23^  apiece? 

Lumber  is  sold  by  the  board  foot.     A  board  foot  is  a  board 
1  ft.  long,  1  ft.  wide,  and  1  in.,  or  less,  thick. 

16.  How  many  board  feet  of  lumber  in  a  board  1  in.  thick, 
1  ft.  wide,  and  5  ft.  long?  the  same  thickness  and  width  and  10  ft. 
long?  12  ft.  long? 

17.  How  many  board  feet  in  a  board  1  in.  thick,  10  ft.  long, 
and  2  ft.  wide?  the  same  thickness  and  length  and  6  in.  wide?  9  in. 
wide?  18  in.  wide?  1  in.  wide?  8  in.  wide? 

18.  How  many  board  feet  in  a  board  12  ft.  long,  4  in.  wide, 
and  1  in.  thick?  2  in.  tbick? 


INTRODUCTION"  13 

19.  How    many  board    feet    in  a    "two  by  four"   scantling 
10  ft.  long  (2"  x  4"  x  10')?  16  ft.  long? 

20.  How  many  board  feet  in  a  "four  by  four"  scantling  10  ft. 
long?     12  ft.  long?  18  ft.  long? 

21.  How  many  board  feet  in  a  fencing  plank  1"  x  6"  x  12'?  in 
5  such  planks? 

22.  How  much  will  the  fencing  lumber  of  problem  12  cost  @ 
$18  per  M  board  feet? 

23.  What  will  be  the  total  cost  of  the  lumber  and  posts  for 
the  fence  on  the  north  and  east  sides  of  the  lot? 

24.  It  cost  8^  apiece  to  have  the  post  holes  bored,  and  it  took 
4  days'  work  by  3  men  @   $1.50  per  day  to  build  the  fence. 
Three  patent  gates  costing  $12  apiece  were  put  in  the  fence. 
Find  the  total  cost  of  the  lot  fence,  including  gates. 

25.  What  was  the  total  cost  of  all  the  fencing  done  on  the 
farm? 

§6.  Areas  of  Fields.  ORAL  WORK 

1.  The  whole  farm  contains  160  acres;   how  many  acres  are 
there  in  the  cornfield?  in  the  wheatfield?  in  the  meadow?   in  the 
south  oatfield?  in  the  pasture?  in  the  lot  around  the  house  and 
barn? 

2.  What  is  the  width  of  each  of  these  fields  in  rods? 

3.  If  the  meadow  were  divided  by  a  north  and  south  central 
line,  how  many  rods  wide  would  each  half  be?     How  many  acres 
would  there  be  in  each  half? 

4..  How  many  square  rods  in  a  strip  1  rd.   wide  and  80  rd. 
long? 

5.  How  many  square  rods  in  an  acre? 

6.  How  wide  must  a  strip  of  land  80  rd.  long  be  to  contain  an 
acre?  40  rd.  long? 

7.  Compare  the  sizes  of  the  wheatfield  and  the  north  oatfield; 
of  the  cornfield  and  the  wheatfield ;  of  the  meadow  and  the  wheat- 
field;  of  the  meadow  and  the  south  oatfield;  of  the  meadow  and 
the  house  and  barn  lots;    of   the  lots  and  the  pasture;    of  the 
pasture  and  the  south  oatfield. 


14  RATIONAL    GRAMMAR    SCHOOL   ARITHMETIC 

WRITTEN  WORK 

1.  What  would  be  the  gross  income  from  the  farm  in  1  yr.  if  it 
were  all  seeded  to  corn  and  the  average  yield  were  47  bu.  per 
acre  and  were  sold  at  28^'  per  bu.?     What  would  be  the  income 
per  acre? 

2.  If  the  farm  were  rented  @  $5  per  acre,  how  much  would 
remain  for  the  tenant  *  from  each  acre?  from  the  entire  farm? 

3.  What  would  be  the  owner's  income  from  the  whole  farm, 
not  allowing  for  expenses? 

4.  Which  is  the  more  profitable  way  for  the  owner  to  rent  his 
farm,  @  $6  per  acre  cash,  or  for  J  of  all  the  crop  delivered  to  mar- 
ket, supposing  that  the  whole  farm  is  planted  to  corn  and  that  a 
yield  of  48  bu.  per  acre  and  a  price  of  30^  per  bushel  can  be 
obtained  every  year? 

NOTE. — Rent  of  the  first  sort  is  called  "cash  rent,"  of  the  second  sort, 
"grain  rent." 

5.  If  this  same  farm  will  produce  20  bu.  of  wheat  per  acre 
and  a  price  of  60^  per  bushel  can  be  obtained  for  it,  which  is  the 
more  profitable  crop  to  the  owner,  wheat  or  corn?  how  much  more 
profitable? 

6.  Compare  the  profits  to  the  owner  of  a  grain-rented  farm 
from  an  oats  crop  of  40  bu.  per  acre  and  a  price  of  22^  per  bushel 
with  the  profits  of  the  wheat  crop  of  problem  5 ;    also  with  corn 
crop  of  problem  4. 

7.  In  problem  5  which  would  bring  the  larger  income  to  the 
owner,  and  how  much,  cash  rent  @  $5  per  A.,  or  grain  rent  @  $ 
delivered? 

8.  Answer  the  same  question  for  the  oats  crop  mentioned  in 
problem  6. 

/       9.  A  cornfield  of  68  acres  produced  an  average  yield  of  48  bu. 
per  acre.     How  many  bushels  did  the  farm  yield? 

10.  It  cost  $5.85  per  acre  to  raise  the  crop.     At  46^  per  bu. 
what  was  the  net  value  of  the  crop? 

11.  If  the  tenant  paid  cash  rent  @  $5.50  per  acre,  what  was 
his  net  profit  from  the  crop?     How  much  did  the  owner  receive? 

*  The  tenant  is  the  farmer  who  raises  the  crop  on  another  man's  farm. 


INTKODUCTKW 


15 


§7.  Habits  of  Animals. 

Certain  pupils  made  observations  of  the  different  sorts  of  beasts, 
birds,  and  insects  which  inhabited  the  neighborhood  of  their  school. 
The  following  table  shows  the  classified  results  of  100  observa- 
tions of  this  kind: 


1 

Migrate  as  winter  approaches 

18 

2 

Store  food  for  winter 

15 

3 

Remain  and  feed  abroad  in  winter 

20 

4 

Hibernate  without  food 

10 

5 

Die  as  winter  comes  on 

27 

6 

Appear  only  in  winter 

10 

Total 

100 

NOTE. — Pupils  are  urged  to  make  their  own  observations,  to  arrange 
them  as  in  the  table,  and  to  use  them  as  indicated  in  the  problems. 

1.  What  fractional  parts  of  the  entire  number  of  different  kinds 
of  animals  do  you  find  in  the  6  classes  separately? 

2.  The  number  of  class  2  equals  what  part  of  the  number  of 
class  1?  of  class  3?   of  class  5? 

3.  How  many  different  kinds  of  animals  per  100  (kinds)  hiber- 
nate without  food? 

4.  What  per  cent  of  all  the  different  kinds  of  animals  of  the 
region  migrate? 

NOTE. — "What  per  cent"  means  "how  many  in  a  hundred"  or  "how 
many  hundred ths." 

5.  What  per  cent  of  the  different  kinds  store  food  for  winter? 

6.  What  part  of  the  number  of  animals  of  the  first  5  classes 
equals  the  number  of  animals  of  class  6? 

7.  The  animals  of  classes  1,  4,  and  5  disappear  from  the  land- 
scape in  winter.     How  many  disappear? 

8.  What  per  cent  of  all  the  animals  in  this  collection  disap- 
pear from  the  winter  landscape? 

9.  The  first  five  classes  appear  in  summer.     Only  classes  2,  3, 
and  6  appear  in  winter.     The  number  of  animals  which  enliven  the 
winter  landscape  equals  what  per  cent  of  the  animals  of  the  sum- 
mer landscape? 


16  RATIONAL    GRAMMAR   SCHOOL    ARITHMETIC 

§8.  Physical  Measurements  (a):  Spirometer. — The  school  should  be 

supplied  with  one  tin  can  or  pail  18  in.  high  and  10  in.  in  diameter, 

and   another  about   9  in.   in  diameter  and 

-  *       * "•   A — a —  9  to  12  in.  in  height.     If  the  school  does 

not   possess   this   apparatus,  have  it   made 
I         L^^^J        I     by  a  tinner.     Fill  the  large  can  nearly  full 
(3  Xfl  0    °^  wa^er  an^  insert   the  small  can,  mouth 

downward,  within  the  water. 

With  an  arrangement  of  pulleys  and 
weights,  w  w,  which  may  be  cans  containing 
sand,  the  inner  can  may  be  counterbalanced 
until  it  moves  easily  either  upward  or  down- 

ward. 

FIGUKB  9  A  rubber  tube  passed  under  the  mouth 

of  the  inverted  can  and  stretched  over  the 

end  of  a  piece  of  glass  tubing,  which  is  held  in  an  upright  position 
inside  the  inverted  can  by  light  cross-braces,  may  be  used  to  con- 
vey air  into  the  can. 

If  the  stopper  at  A  is  drawn,  the  can  sinks  readily  in  the 
water;  after  which  the  stopper  is  inserted  air-tight. 

A  foot  rule,  graduated  to  eighths  or  sixteenths  of  an  inch, 
stuck  with  putty  to  the  side  of  the  inside  can,  permits  the  read- 
ings to  be  taken. 

Two  pails  of  different  sizes  may  serve  as  a  very  satisfactory  sub- 
stitute for  the  cans.  The  method  of  fitting  them  up  for  use  is 
plain  from  Fig.  9. 

1.  If  the  inside  diameter  of  the  inverted   can  is  8.8  in.,  for 
each  inch  the  inner  can  sinks  into  the  water,  there  will  be  60.85 
cu.  in.  of  water  inside  the  can,  the  stopper  at  A  being  removed. 
How  many  cubic  inches  of  water  will  there  be  in  the  can  when  it 
has  sunk  2  inches?  5  inches?  8  inches? 

NOTE. — If  a  pail  is  used,  the  corresponding  number  of  cubic  inches 
is  found  by  measuring  the  distance  in  inches  across  the  mouth  of  the  pail, 
multiplying  one-half  this  distance  by  itself,  and  the  product  by  3f. 

2.  How  many  cubic  inches  of  water  will  there  be  in  the  can 
when  it  is  sunk  in  the  water  only  1J  inches?  If  inches?  2-J-  inches? 
2TV  inches? 

3.  Let  the  can  now  be  pushed  down  as  far  as  it  will  go  and  the 
stopper  at  A  be  pushed  in  tightly.     Read  the  scale.     When  a  pupil 
blows  through  the  tube  until  the  inside  can  rises  1  inch,  how 


INTRODUCTION  17 

many  cubic  inches  of  air  has  he  expelled  from  his  lungs  into  the 
can?  How  many  when  the  scale  indicates  that  the  can  has  risen 
I  inch?  1TV  inches?  1J  inches?  lT5g-  inches?  If  inches? 

4.  Fill  your  lungs  full  and  blow  into  the  tube  as  long  as  you 
can  without  danger.     What  is  the  capacity  of  your  lungs  in  cubic 
inches? 

5.  The  average  lung  capacity  for  a  man  5  ft.  8  in.  tall  is  204 
cu.   in.     This    average  capacity  may  be  called  the   normal  lung 
capacity.     How  much  does  your  lung  capacity  fall  short  of  the 
normal  value  for  a  man? 

6.  The  normal  lung  capacity  in  cubic  inches  for  a  man  is  3 
times  his  height  in  inches.     For  a  woman,  it  is  2.6  times  the 
height.     Divide  your  lung  capacity  by  your  height  in  inches,  and 
compare  your  quotient  with  these  numbers. 

7.  The  chest  measure  of  a  man  should  be  not  less  than  half  his 
height.     How  much  greater  or  less  is  your  chest  measure  than  it 
should  be  according  to  this  law? 

§9.  Physical  Measurements  (b). 

1.  Fill  out  in  your  notebook  a  record  like  the  one  below,  which 
is  a  copy  from  a  student's  notebook : 

Name,  John  Morrow;  age,  10  yr.;  height,  52  in.;  weight,  61  Ib. 

(  Inspiration,  28    in. 
Chest  measures,  •<  Expiration,  2Jfo  in.     Lung  capacity,  104  cu.  in. 

(  Mean,*          26\  in. 

Height  in  inches  by  3     (if  a  boy).     Result  52  X  3  =  156  cu.  in. 
Height  in  inches  by  2.6  (if  a  girl). 

Average  to  each  inch  of  height,  2  cu.  in. ,  for  104  -*-  52  =  2. 
Chest  average  =  S6\  in.     Half  height  =  26  in. 

2.  How    much  do  your    measures  exceed  or  fall  short  of  the 
normal  values? 

3.  Do  your  measures  show  that  you  need  chest  exercise? 

4.  How  many  in  the  class  or  room  are  above  or  below  the 
normal? 


wan  of ;:  numbers  is  half  their  sum. 


18 


RATIONAL    GRAMMAR    SCHOOL    ARITHMETIC 


5.  Compare  class  or  room  averages  to  see  whether  the  average 
for  class  or  room  is  about  normal. 

6.  Make  the  comparison  of  problem  5  after  a  course  of  exercises 
in  physical  training,  and  record  the  changes  due  to  the  exercises. 

7.  Every  month  or  two  during  the  year  make  the  computa- 
tions called  for  in  problems  2  and  5 ;  keep  your  results,  and  show 
what  changes  are  taking  place  in  your  measures. 


§10.  Physical  Measurements  (c). — The  following  table  gives  the 
average  height  in  inches  and  the  average  weight  in  pounds  for  boys 
and  for  girls  year  by  year  from  4  to  15  years  of  age.  The  num- 
bers are  the  averages  of  careful  measures  of  heights  and  weights  of 
hundreds  of  boys  and  girls  in  the  schools  of  Chicago,  Boston, 
Cincinnati,  and  St.  Louis.  These  averages  may  be  called  normal 
values  for  boys  and  girls  of  school  age : 


HEIGHT 

YEARLY  GROWTH 

WEIGHT 

YEARLY  GROWTH 

AGE 

Boys 

Girls 

Boys 

Girls 

Boys 

Girls 

Boys 

Girls 

4yr. 

39.2  in. 

39.0  in. 

37.31b. 

35.31b. 

5 

41.6 

41.4 

40.6 

39.7 

6 

43.7 

43.3 

44.7 

43.0 

7 

45.8 

45.5 

48.7 

47.0 

8 

47.9 

47.6 

53.8 

52.0 

9 

49.7 

49.5 

58.7 

57.1 

10 

51.6 

51.3 

64.8 

62.2 

11 

53.5 

53.4 

70.1 

68.1 

12 

55.2 

55.8 

76.7 

77.4 

13 

57.3 

58.5 

84.9 

88.4 

14 

60.0 

60.2 

95.0 

98.3 

15 

62.4 

61.3 

106.5 

105.0 

1.  Subtract  each  number  in  the  column  of  heights  from  the 
number  next  below  it  and  find  the  growth  in  height  for  boys  and 
for  girls  from  year  to  year.     Do  the  same  for  the  weights. 

2.  When  is  the  yearly  growth  in  height  greatest  for  boys? 
for  girls?     What  is  the  yearly  growth  in  each  case? 

3.  Make  and  solve  similar  problems  for  weights. 

4.  Compare  your  own  height   and   weight   with    the  normal 
values  for  your  own  age.     How  much  do  you  exceed  or  fall  short 
of  the  average? 


INTRODUCTION  19 

5.  How  do  the  averages  for  your  room  compare  with  the  values 
in  the  table  for  the  same  years? 

6.  How  much  do  your  height  and  weight  exceed  or  fall  short 
of  the  averages  for  pupils  of  your  own  age  in  your  room? 

§11.  Vital  Statistics.— The  New  York  Board  of  Health  gwes  the 
annual  death  rate  per  thousand  of  population  as  follows : 


1886 26.0 

1887 26.3 

1888 26.4 

1889 25.3 

1890 24.9 

1891..  ..26.3 


1892 25.9 

1893 25.3 

1894 22.8 

1895 23.1 

1896  (1st  half  year)  21. 5 


1.  What  is  the  average  death  rate  for  1886-89? 

NOTE. — The  average  death  rate  for  4  years  is  \  of  the  sum  of  the  rates 
for  the  single  years. 

2.  What  for  the  years  1890-93? 

3.  What  is  it  for  the  years  1894-96,  assuming  that  the  death 
rate  of  the  first  half  will  be  the  rate  for  the  entire  year? 

4.  In  1894  special  efforts  were  directed  toward  street  cleaning 
and  better  city  housekeeping  generally.     What  reduction  in  death 
rate  was  made  in  the  one  year  from  1893  to  1894  as  a  consequence? 

5.  If  the  population  of  New  York  City  for  1893-94  be  taken 
as  2,800,000,  the  lives  of  how  many  people  were  saved  by  the 
improved  conditions  during  1894? 

6.  Compare  the  death  rate  of  your  own  town,  city,  or  county 
with  the  values  of  this  table.     How  much  does  it  exceed  or  fall 
short  of  the  largest  'value  of  the  table?   the  smallest  value?   the 
average  value? 

§12.  Wind  Pressure. — The  velocity  of  wind  in  miles  per  hour  and 
the  pressure  per  square  foot  are  as  given  here : 


Light  breeze 3|  mi.       .75  oz. 

Moderate  breeze  . .  6£   "       3.33   " 

Fresh  breeze 16^   "    lib.  5   " 

Stiff  breeze 32i  "    5  "  3   " 


Strong  gale 56|  mi.  15  Ib.  9oz. 

Hurricane 79^  "    31  "  4  " 

Violent  hurricane  97£  "    46  "  12  " 


NOTE. — Use  measures  from  your  own  schoolhouse  or  from  other  build- 
ings in  your  neighborhood  in  preference  to  the  numbers  in  the  problems. 


20 


RATIONAL   GRAMMAR   SCHOOL   ARITHMETIC 


FIGURE  10 


1.  What  is  the  total  pressure  in  pounds  on  the 
west  side  of  a  house  30  ft.  by  25  ft.,  due  to  a  light 
breeze  from  the  west?  to  a  stiff  breeze?  to  a  strong 
gale? 

2.  What  is  the  pressure  on  the  side  of  a  tall 
building  265  ft.  by  80  ft.,  due  to  a  hurricane  blow- 
ing squarely  against  it? 

3.  Find  the  wind  pressure  against  the  side  of 
a  load  of  hay  25  ft.  long  and  12  ft.  high,  due  to  a 
strong  gale  blowing  squarely  against  it. 

4.  Find  the  number  of  pounds  pressure  against 
a  signboard  fence  100  ft.  by  28  ft.,  due  to  a  strong 
gale  blowing  squarely  against  it. 

5.  What  is  the  wind  pressure  tending  to  over- 
turn a  square  chimney  30  ft.  wide  at  the  base, 
15  ft.  wide  at  the  top,  and  175  ft.  high,  due  to  a 
violent  hurricane  blowing  souarely  against  one  of 
the  flat  faces? 

NOTE. — To  obtain  the  area  of  the  effective  surface 
against  which  the  wind  is  blowing,  multiply  the  height 
of  the  chimney  by  its  breadth  halfway  up.  This  breadth 
is  the  half  sum  of  the  widths  at  the  top  and  bottom 
(Fig.  10). 

6.  Find    the  wind 
pressure  on  the  side  of 

to    a    violent   hurricane    — * 


a  tree,    due 

blowing  against  the  triangular  top, 

ft.    across   the  base   and   43   ft.    high 

(see  Fig.  11). 

7.  Find  the  wind  pressure  on  the 
side  of  a  passenger  coach  70  ft.  long 
and  15  ft.  high,  due  to  a  stiff  breeze 
blowing  squarely  against  it. 

(a)  Suppose  the  wind  to  blow  against 
the  whole  rectangle  15'x70';  then, 

(b)  Suppose  the  wind  is  obstructed  by  only  -J- 
4V  x  70',  from  the  track  to  the  bottom 
running  the  length  of  the  coach. 


FIGURE  11 


of   the  strip 
of  the  coach  box  and 


INTRODUCTION 


§13.   Dairying. — The  following  table  is  an  extract  from  the  milk 
record  of  the  University  of  Illinois  herd  of  milk-cows : 

WEIGHT  OF  MORNING  (A.M.)  AND  EVENING  (P.M.)  MTLKINGS  IN  POUNDS 


1901 

QUEEN 

BEAUTY 

BEECH- 
WOOD 

MYRTLE 

SPOT 

ROSE 

IV  rw 

TOTAL 

i"  Of 

a.m. 

p.m. 

a.m. 

p.m. 

a.m. 

p.m. 

a.m. 

p.m. 

a.m. 

p.m. 

a.m. 

p.m. 

1 

6.5 

6.5 

11.0 

10.4 

12.5 

10.2 

7.3 

5.7 

12.0 

11.5 

11.0 

9.6 

5.6 

6.0 

11.0 

8.4 

11.0 

11.0 

7.2 

5.6 

11.5 

10.5 

11.3 

8.8 

6.4 

6.0 

10.8 

9.5 

11.7 

11.5 

7.3 

5.7 

12.0 

10.4 

11.2 

9.8 

6.0 

6.5 

12.5 

10.2 

12.6 

13.0 

7.2 

6.0 

11.5 

11.5 

11.0 

9.8 

6.0 

7.5 

11.4 

10.2 

12.0 

11.4 

6.7 

5.8 

11.6 

11.3 

11.0 

10.2 

e 

6.6 

6.8 

12.0 

10.0 

8.0 

11.6 

8.0 

5.0 

12.5 

11.8 

11.7 

10.0 

7 

7.4 

8.5 

10.8 

12.4 

12.0 

11.7 

7.0 

7.2 

12.0 

12.5 

11.6 

12.0 

8 

7.8 

7.3 

11.2 

11.6 

13.2 

12.2 

7.2 

5.7 

11.5 

11.3 

11.5 

10.1 

9 

7.5 

6.4 

11.6 

10.4 

12.0 

11.6 

6.6 

5.4 

12.0 

12.2 

10.7 

100 

10 

7.8 

7.2 

11.0 

10.4 

12.0 

12.0 

6.7 

4.7 

11.5 

8.5 

11.8 

10.6 

11 

6.8 

7.3 

12.2 

11.0 

13.6 

11.5 

7.7 

5.3 

9.0 

10.0 

12.3 

10.0 

12 

7.8 

7.7 

12.5 

12.0 

13.0 

11.6 

8.0 

5.5 

12.0 

12.6 

12.2 

10.6 

13 

7.6 

8.0 

13.4 

11.5 

13.5 

13.0 

7.2 

5.8 

11.1 

13.0 

11.8 

112 

14 

7.0 

8.0 

12.0 

11.8 

14.0 

12.7 

7.7 

5.7 

12.5 

12.7 

11.5 

10.3 

15 

8.3 

7.5 

13.0 

10.7 

15.0 

12.0 

7.6 

5.0 

14.0 

12.4 

12.3 

9.0 

16 

9.0 

7.0 

12.5 

11.0 

15.2 

13.7 

7.0 

5.5 

12.7 

12.3 

14.1 

8.5 

17 

8.0 

7.0 

12.5 

10.5 

14.0 

13.0 

6.1 

6.4 

10.0 

9.0 

12.0 

10.5 

18 

8.1 

7.7 

13.4 

10.9 

14.0 

11.5 

6.6 

5.0 

12.5 

11.5 

11.0 

9.1 

19 

7.0 

7.0 

11.5 

11.4 

13.6 

11.0 

6.2 

5.3 

12  2 

11.7 

11.1 

10.0 

20 

7.5 

6.5 

12.6 

10.5 

12.7 

12.0 

6.5 

4.9 

1L8 

11.4 

11.5 

9.5 

21 

7.0 

7.6 

12.0 

10.2 

13.7 

11.3 

6.1 

5.2 

13.0 

11.0 

10.6 

9.6 

22 

8.0 

7.5 

12.2 

11.9 

14.7 

11.2 

6.7 

6.0 

12.4 

10.7 

11.3 

9.5 

23 

7.5 

9.0 

12.2 

11.0 

14.4 

12.0 

7.0 

5.5 

12.8 

10.8 

11.8 

10.4 

24 

7.0 

7.1 

13.5 

10.6 

13.8 

11.5 

7.2 

5.1 

13.0 

10.7 

12.0 

10.4 

25 

7.4 

8.0 

12.5 

11.2 

14.0 

10.5 

7.4 

5.7 

12.7 

11.7 

12.0 

11.0 

26 

6.5 

8.4 

12.0 

11.8 

12.0 

12.0 

6.1 

5.8 

13.2 

11.5 

11.5 

10.5 

27 

7.5 

8.5 

13.7 

11.8 

14.0 

13.0 

6.1 

5.8 

13.0 

12.7 

11.4 

10.5 

28 

7.2 

9.2 

12.0 

12.0 

13.6 

13.8 

6.6 

4.9 

13.1 

12.5 

11.7 

10.5 

29 

6.5 

9.0 

13.5 

12  6 

15.0 

13.5 

6.7 

5.0 

10.2 

12.6 

12.5 

12.6 

30 

7.2 

8.0 

11.7 

12.7 

14.2 

12.2 

6.0 

4.3 

12.4 

11.5 

12.7 

10.3 

1.  How  many  pounds  of  milk  did  Queen  give  the  first  week  of 
Nov.,  1901,  at  the  morning  milkings?  at  the  evening  milkings? 
at  both  milkings? 

2.  At  which  milking  did  Queen  give  most  milk  and  how  much 
more  than  at  the  smallest  milking? 

3.  Make  and  answer  similar  questions  for  the  remaining  weeks 
and  for  any  or  all  of  the  6  cows. 


22 


KATIOKAL    GRAMMAR    SCHOOL   ARITHMETIC 


4.  What  was  the  total  weight  of  milk  at  the  morning  milkings 
for  the  whole  herd  on  Nov.  1st?  at  the  evening  milkings  on  the 
same  date?  at  both  milkings?     Put  the  answer  to  the  last  question 
in  the  column  of  totals  at  the  right. 

5.  Make  and  answer  such  questions  as  4,  for  other  dates  and 
for  each  week  of  November. 

6.  If  a  quart  of  milk  weighs  2.2  lb.,   what  is  the  morning 
milking  of  the  herd  worth  at  6±$  per  quart  for  Nov.  1?  the  even- 
ing milking? 

7.  What  is  the  milk  of  the  herd  worth  each  week  if  milk  sells 
at  6^0  per  quart?  what  for  the  month? 

8.  Make  such  problems  as  7,  for  single  cows  and  solve  your 
own  problems. 

9.  Has  there  been  any  gain,  or  loss,  in  the  weekly  milk  yield 
of  the  herd,  or  of   individual  cows,   during  the  month?     If  so, 
how  much? 

§14.  The  Thermometer. — The  official  record  of  hourly  temper- 
atures from  6  p.m.  February  27  to  6  p.m.  March  1,  1902,  as  given 
in  a  daily  paper,  is  as  follows : 


FEB.  27;  NIGHT 

FEB.  28;  DAY 

FEB.  28  ;  NIGHT 

MARCH  1;  DAY 

6pm              38° 

Gam              40° 

6pm              39° 

6am              34C 

7  p.  m  38 

7  a.  m  41 

7  p.  m  38 

7  a.  m              34 

8pm.             37 

8am              41 

8pm              37 

8am              33 

9pm.             37 

9am              41 

9pm              37 

9am               33 

10  p  m.             39 

10  a.  m              43 

10  p   m              37 

10  a  m              32 

11  p  m           .39 

11  a  m              44 

11  p  m              37 

11  a  m              32 

12  midnight     38 

12  m                  38 

12  midnight     36 

12  m                  32 

Average.  ..... 

Average  

1  a.  m  36 

1  p.  m  31 

12  midnight     38 

12  m                  38 

2am              35 

2pm              31 

1  a.  m  38 

1pm              35 

3am              34 

3  p.  m              30 

2am.        .38 

2pm              35 

4  a.  m           .  .  34 

4  p.  m              29 

3  a.  m  38 

3  p.  m               37 

5  a.  m   .        .  .  34 

5pm              28 

4  a.  m  37 

4  p  m.             39 

6am           .  .  34 

6  p.  m.              28 

5  a.  m  39 

5  p.  m           .  .  40 

Sum            .... 

Sum  . 

6  a.  m  40 

6  p.  m  39 

Average  

Average.  . 

Average.  ..... 

Average  

NOTE. — The  average  of  temperatures,  or  numbers,  is  found  by  adding 
them  and  dividing  the  sum  by  the  number  of  temperatures,  or  numbers, 
which  have  been  added.  To  average  means  to  find  the  average. 


INTRODUCTION 


80-: 


60-: 
5CK 


1CH 

|zo-i 
30-: 


1.  Average   the    temperatures    on    Feb.   27  from    6    p.m.  to 
midnight;    from  midnight  to  6  a.m. 

2.  What  was  the  average  temperature  on  Feb.  28 
from  6  a.m.  to  12  m.?  from  12  m.  to  6  p.m.? 

3.  What  is  the  difference  between  the  two  aver- 
ages of  problem  1?  of  problem  2?     What  do  these 
differences  mean? 

4.  What  is  the  difference  between  the  lowest  and 
highest  temperatures  from  6  p.m.  Feb.  27  to  6  p.m. 
Feb.  28?     This  is  called  the  range  of  temperature 
for  the  day. 

5.  The  average  temperature  for  Feb.  for  30  yr.  is 
26°.     How  much  does  the  average  temperature  for 
the  24  hr.  following  6  p.m.  Feb.  27  exceed  the  thirty- 
year  average? 

6.  Find  the  average  temperature  for  the  night 
of  Feb.  28;  for  the  day  of  March  1. 

7.  Find  the  difference  between  these  averages. 
What  does  this  difference  show? 

8.  Make  and  solve  such  problems  from  outdoor 
thermometer  readings  at  your  schoolhouse. 

9.  What  is  the  reading  of  the  thermometer  of  Fig.  12?     What 
would  the  thermometer  read  if  the  top  of  the  mercury  column 
stood  at  "freezing"?  at  "summer  heat"?  at  "blood  heat"? 

10.  Average   the   thermometer   readings  taken  at   your  own 
school  for  each  hour  from  9  a.m.  to  4  p.m.  for  a  month,  and  keep 
in  a  notebook  a  record  of  your  readings  and  averages. 

11.  Lay  off 
on  cross-  lined 
paper  the  read- 
ings of  problem 
10  as  the  read- 
ings of  the  table 
from  9  a.m.  to  4 
p.m.  of  Feb.  28 
are  laid  off  in  Fig.  13;  draw  a  smooth  curve,  free-hand,  through 
the  points.  Such  a  curve  gives  a  picture  of  temperature  changes. 


FlGUKE  12 


50° 
40° 
30" 
20° 
10° 
0°, 


9      10     11     12     1       2      3      4 

A.M.  A.M.  A.M.   M.  P.M.  P.M.  P.M.  P.M. 

February  28 
FIGURE  13 


9      10     11      12      1       234 

P.M.   P.M.  P.M.     M.   A.M.  A.M.  A.M.  A.M. 

February  28— March  l 
FIGURE  14 


24  RATIONAL    GRAMMAR    SCHOOL    ARITHMETIC 

12.  From  a  daily  paper  obtain  the  hourly  readings  from  9  p.m. 
to  4  a.m.  and  lay  them  off  to  scale  as  the  readings  of  the  table 
from  9  p.m.  Feb.  28  to  4  a.m.  March  1  are  laid  off  in  Fig.  14. 
Draw  the  free-hand  curve  as  before  and  find  whether  the  day  or 
night  temperatures  are  the  steadier. 

13.  Average  your  readings  from  9  a.m.  to  4  p.m.  and  draw  a 
straight,  horizontal    line  on  the   diagram  of    your   temperature 
curve,  at  a  distance  above  the  horizontal  zero  line  equal  to  the 
average,  as  the  line  av  of  Fig.  13  is  drawn.     This  is  the  average 
line  for  these  8  hours. 


45° 

40 


10C 
5°- 


910H121       23456789     10     11     12     1       23456789 
A.M.  A.M,A.M».MI  P.M.  P.M. P.M.  P.M.  P.M.  P.M.  P.M.  P.M.  P.M.  P.M.  P.M.    M.  A.M.  A.M.  A.M.  A.M.  A.M. A.M.  A.M.  A.M.  A.M. 

Changes  from  9  a.m.  February  28,  to  9  a.m.  March  1 
FIGURE  15 

14.  From  a  daily  paper,  or  by  other  means,  obtain  the  hourly 
readings  for  24  hr.  in  succession.     Draw  the  curve  for  them  as 
the  curve  of  Fig.   15  is  drawn  for  the  readings  from  9  a.m.  Feb. 
28  to  9  a.m.  March  1. 

15.  Lay  oft  to  scale  the  noon   temperatures  from  day  to  day 
for  a  month  on  a  piece  of  cross-ruled  paper,  and  draw  as  smooth 
a  free-hand  line  as  possible,  uniting  the  points.     This  curve  show? 
the  changes  of  noon  temperatures  for  the  whole  month. 

16.  Lay  off  the  averages,  as  in  problem  13,  for  each  day  for  a 
week  and  obtain  the  changes  of  temperature  for  this  week. 

17.  Extend  the  curve  of  problem  16  for  a  month. 

18.  In  the  same  way  draw  a  curve  through  points  obtained  bj 
laying  down  to  scale  the  average  values  for  the  12  mo.  of  any 


INTRODUCTION" 


year  since  1872.  The  vertical  lines  should  correspond  to  months 
of  the  year.  (See  Chicago  Weather  Bureau  Summary.) 

19.  Draw  a  curve  for  the  monthly  averages  for  the  29  years. 

§15.  November  Meteors. — The  earth  in  its  yearly  journey  around 
the  sun  passed  through  a  swarm  of  meteors  Nov.  13-15, 
1901.  The  meteors  were  seen  in  great  numbers  as  shooting  stars 
by  observers  at  different  places,  and  were  counted  and  recorded 
as  in  the  table  below. 

1.  The  swarm  was  so  large  that  it  took  the  earth,  JEJ,  which  moves 
18.6  mi.  per  second,  48  hr.  .  pati, 
to  move  across  it,  from  a  to 

b.     How    long  is  aft? 

2.  The     meteors     were 
thicker  at  some  places  than 
at   others  and  the  greatest 
number  of  shooting  stars  was 
seen  when   the    earth    was 
where    the    meteors     were 

thickest.  From  the  table  below,  on  which  day  did  the  earth  go 
through  the  densest  part  of  the  swarm? 

COUNTS  OP  SHOOTING  STARS,  NOVEMBER,  1901 


FIGURE  16 


HOUR 

A.M. 

NUMBER  SEEN 
Nov.  13 
AT  CLAREMONT, 
CAL. 

NUMBER  SEEN  NOVEMBER  14 

NUMBER  SEEN 
Nov.  15 
AT  MINNEAPOLIS 

AT  CLAREMONT, 
CAT,. 

AT  MINNE- 
APOLIS 

12  to  1 

19 

51 

5 

5 

1  to  2 

27 

85 

19 

28 

2  to  3 

20 

95 

54 

35 

3  to  4 

22 

238 

62 

,    87 

4  to  5 

30 

682 

106 

19 

5  to  5:30 

19 

306 

132 

11 

TOTALS 

3.  Find  the  whole  number  of  shooting  stars  counted  at  Minne- 
apolis on  Nov.  14;  at  Claremont. 

4.  Answer  the  same  question  for  Nov.  15  for  Minneapolis,  and 
for  Nov.  13  for  Claremont. 


RATIONAL   GRAMMAR   SCHOOL   ARITHMETIC 


§16.  The  Barometer. — Close  the  upper  end  of  a  glass  tube  40 
inches  long  and  -f-$  of  an  inch,  or  more,  in  diameter,  by  heating  it 
in  a  flame  until  the  glass  is  soft,  and  drawing  off  a 
small  piece  near  the  top  by  a  gradual  stretch.  Draw 
the  end  of  a  piece  of  rubber  tube  8  or  10  in.  long 
over  the  lower  end  of  the  glass  tube.  Insert  into 
the  other  end  of  the  rubber  tube  a  piece  of  glass  tube 
about  G  inches  long  and  open  at  both  ends.  Straighten 
out  the  entire  tube  and  pour  into  it  some  44  in.  of  mer- 
cury. Bend  the  rubber  tube  and  tie  the  whole  to  a 
board  by  a  string  or  wire,  as  shown  in  Fig.  17,  with 
the  mouth  of  the  short  glass  tube  upward. 

Provide  a  home-made  scale  in  inches  and  tenths 
(use  the  scale  of  tenths  of  Pig.  3)  at  the  top  and  bottom 
of  the  board,  extending  a  short  distance  above  and  below 
the  tops  of  the  mercury  columns.  The  graduations 
should  be  numbered  in  inches  above  the  same  hori- 
zontal line,  as  ab,  across  the  lower  end  of  the  board. 
The  scale  at  the  right  gives  the  length  of  the  long 
column  and  that  at  the  left  the  length  of  the  short 
column.  The  difference  of  these  lengths  is  the  length 
of  the  mercury  column  which  is  supported  by  the  air 
through  the  mouth  of  the  short  tube. 


NOTE. — If  desired,  these  differences  may  be  obtained  by 
measuring  with  a  yard-stick  the  lengths  ad  and  be  and  sub- 
tracting. 


FIG.  17 


Nov.  11. 

Long  col.  33.15ft. 
Short  col.  2.75  ft. 

Diff.... 30.40  ft. 

MARIAN  JOHNSON. 


With  an  apparatus  like  this,  a  fourth 
grade  class  measured  the  two  columns  on 
8  successive  days,  and  found  the  difference 
between  their  lengths.  This  difference  be- 
tween the  lengths  of  the  mercury  columns  is 
called  the  "reading"  of  the  barometer. 
The  readings  taken  were  laid  down  to 
scale  on  the  vertical  lines  of  a  drawing, 
as  shown  in  Fig.  18,  and  a  smooth  curve  was 
then  drawn  free-hand  through  the  points.  This  curve  shows  to 
the  eye  how  the  noon  heights  of  the  mercury  columns  varied  from 
Nov.  11  to  Nov.  20. 

1.  Can  you  tell  from  the  words  "clear,  cloudy"  in  Fig.  18, 
what  kind  of  weather  follows  a  high  barometer,  a  low  barometer, 
a  falling  barometer,  a  rising  barometer  9 


INTRODUCTION 


O.b  ' 
0.0" 
9.5" 

3 

\ 

s~ 

N 

\ 

\ 

/ 

V 

"^^^ 

•  

xZ 

_-/ 

ON              NOON             NOON              NOON             NOON               NOON              NOON             NOON 
0.4             29.8           29.35           29.8            30.2             29.8             29.7            29.65 

M.J.              R.W.            F.C.              M.F.              J.I.              R.B.            J.M.              R.Q. 

FIGURE  18 

barometer  reading  en- 
able one  to  predict 
what  sort  of  weather 
we  are  likely  to  have?  29.5 

NOTE.  —  The  pupils 
whose  initials  stand  be- 
low the  readings  were 
the  readers  and  record- 
ers of  the  measured  dif- 
ferences. 
TABLE  OF  HIGHEST,  LOWEST,  AND  MEAN  BAROMETER  READINGS  IN  INCHES 


1900 

HIGHEST 

LOWEST 

MEAN 

1901 

HIGHEST 

LOWEST 

MEAN 

Oct. 
Nov. 
Dec. 

30.48 
30.61 
30.52 

29.46 
29.60 
29.37 

Oct. 
Nov. 
Dec. 

30.40 
30.58 
30.52 

29.58 
29.63 
29.60 

Autumn  Averaj 
Autumn  Range 

ye  .  . 

Autumn  Average  
Autumn  Range  

1901 

HIGHEST 

LOWEST 

MEAN 

1902 

HIGHEST 

LOWEST 

MEAN 

Jan. 
Feb. 
Mar. 

30.72 
30.61 

30.28 

29.45 
29.63 
29.41 

Jan. 
Feb. 
Mar. 

30.94 
30.51 
30.58 

29.71 

28.73 
29.11 

Winte 
Winte 

r  Average 
r  Range 

5   

Winte 
Winte 

r  Average 
r  Range 

1901 

HIGHEST 

LOWEST 

MEAN 

1902 

HIGHEST 

LOWEST 

MEAN 

Apr. 
May 

June 

30.51 
30.30 
30.18 

29.36 
29.55 
29.71 

Apr. 
May 
June 

30.31 
30.35 
30.20 

29.25 
29.68 
29.37 

Spring 
Spring 

Average  
Range  ... 

Spring 
Spring 

Average 
1  Ran  ire 

1901 

HIGHEST 

LOWEST 

MEAN 

1902 

HIGHEST 

LOWEST 

MEAN 

July 
Aug. 
Sept. 

30.18 
30.20 
30.34 

29.68 
29.81 
29.58 

July 
Aug. 
Sept. 

30.19 
30.23 
30.27 

29.66 
29.75 
29.56 

Summer  Averag 
Summer  Ranee 

'Q 

Summer  Averag 
Summer  Ranee. 

e  

28  RATIONAL   GRAMMAR    SCHOOL   ARITHMETIC 

3.  Find  the  mean  reading  by  taking  £  the  sum  of  the  highest 
and  lowest  readings  for  October,  1900;  for  November;  for  Decem- 
ber. 

4.  Average  these  means  and  write  the  average  in  the  means 
column  opposite  "Autumn  Average." 

5.  Treat  all  8  parts  of  the  table  in  the  same  way. 

6.  What  is  the  range  (difference  between  the  greatest  and  least 
readings)  of  the  barometer  during  the  autumn  of  1900?     When 
did  the  greatest  reading  occur?  the  least?  , 

7.  Answer  questions  similar  to  4  for  each  of  the  8  seasons  of 
the  table. 

8.  For  which  season  of  the  year  is  the  average  barometric  height 
greatest?  least?     What  is  the  range  for  each  of  the  2  years? 

§17.  Farm  Account  Keeping. — The  forms  below  show  how  a  cer- 
tain farmer  keeps  a  systematic  account  of  receipts  and  expendi- 
tures with  each  of  his  fields.  The  fields  referred  to  are  those 
shown  in  the  drawing  of  Fig.  7,  p.  11. 


WRITTEN"   WORK 

The  items  for  which  money  was  paid  out  or  received  were 
put  down  with  the  dates  of  payment  or  of  receipt  in  a  day-book 
thus: 

Day-Book  for  Forty- Acre  Cornfield 

1.  Paid  for  4  da.  work  removing  stalks  @  §1.15,  Mar.  18. 

2.  Paid  for  9  da.  plowing  @  §1.15,  Apr.  30. 

3.  Paid  for  2  da.  harrowing  @  §1.00,  May  1. 

4.  Paid  for  3fc  da.  planting  corn  @  §1.20,  May  18. 

5.  Paid  for  6  bu.  seed  corn  @  90^,  May  18. 

6.  Paid  for  2  da.  replanting  corn  @  §1.15,  May  30. 

7.  Paid  for  5  da.  harrowing  corn  @  §1.15,  May  30. 

8.  Paid  for  7  da.  cultivating  @  §1.15,  June  15. 

9.  Paid  for  6  da.  cultivating  @  §1.15,  June  30. 

10.  Paid  for  6  da.  cultivating  @  §1.15,  July  15. 

11.  Paid  for  cutting  240  shocks  corn  @  8^,  Sept.  15. 

12.  Paid  for  husking  1392  bu.  corn  @  3^,  Nov.  20. 

13.  Paid  for  shelling  1336  bu.  corn  @  f 0,  Jan.  10. 

14.  Sold  1336  bu.  corn  @  38^,  Jan.  10. 

15.  Received  pay  for  3  mo.  pasture  @  §1.50,  Feb.  28. 

1.  These  items  are  here  arranged  in  the  form  of  a  receipt  and 
expenditure  account.  Fill  out  the  vacant  columns,  and  find  the 
totals  and  the  net  profit  for  this  forty-acre  cornfield: 


INTRODUCTION 


In  Account  with  Forty-Acre  Cornfield 
EXPENDITURES  RECEIPTS 


Mar.  18 

Removing  stalks,  4  da.,  @ 

Jan.  10 

1  336  bu.  corn®,  380 

$1.15 

Feb.  28 

3  mo.  pasture  @  $1.50 

Apr.  30 

Plowing,  9  da.,  @$1.15 

May    1 

Harrowing,  2  da.,  @  $1.00 

TOTAL 

"     18 

Planting  corn,  3yz  da.,   @ 

$1.20 

Expenditures     to    be    de- 

"    18 

Seed  corn,  6bu.,@  900 

ducted 

"      30 

"      30 

Replanting,  2  da.,  @  $1.15 
Harrowing  corn,  5  da.,  @ 

Net  profit  from  40  acres 
"        "       per  acre 

$1.15 

June  1  5 
••     30 

Cultivating,  7  da.,   @  $1.15 
6  da.,   @$1.15 

Account    closed    March    1, 

July  15 

6  da.,   @  $1.15 

1901 

Sept.  15 

Ciittiiig  240  shocks  corn  @, 

8^0 

Nov/20 

Husking  1392  bu.  corn  @  30 

Jan.  10 

Shelling  1336  bu.  corn  @. 
X* 

TOTAL 

2.  From  the  following  list  of  day-book  items  for  the  twenty- 
acre  wheatfield  the  account  below  is  arranged.  Fill  out  the  vacant 
columns  and  find  the  totals,  the  net  profit  on  the  whole  field,  and 
the  net  profit  per  acre. 

Day-Bookfor  Twenty- Acre  Wheatfield 

1.  Paid  for  4  da.  plowing  @  12.50,  Oct.  20. 

2.  Paid  for  3  da.  harrowing  and  rolling  @  $2.25,  Oct.  25. 

3.  Paid  for  25  bu.  seed  wheat  @  $1.25,  Oct.  25. 

4.  Paid  for  2£  da.  drilling  wheat  @  $2.50,  Oct.  30. 

5.  Paid  for  cutting  20  acres  wheat  @  75^,  July  15. 

6.  Paid  for  2  da.  shocking  wheat  @  $1.75,  July  15. 

7.  Paid  for  threshing  480  bu.  wheat  @  6/-,  Aug.  30. 

8.  Paid  for  3£  da.  help  in  threshing  @  $1.50,  Aug.  30. 

9.  Sold  425  bu.  wheat  @  68/,  Dec.  22. 

10.  Received  pay  for  4  mo.  pasture  @  $1.50,  Mar.  2. 

In  Account  with  Twenty-Acre  Wheatfield 
EXPENDITURES  RECEIPTS 


Oct.    20 

Plowing,  4  da.,  ©  $2.50 

Dec.  22 

Wheat,  425  bu.,  @  680 

"      25 

Harrowing    and    rolling, 

Mar.    2 

Pasture,  4  mo.,  @  $1.50 

3  da.,  @$2.25 

"      25 

Seed  wheat,  25  bu.,  @  $1.25 

Expenditures    to    be    de- 

"     30 
July  15 

Drilling,  2yz  da.,  @  $2.50 
Cutting  20  acres  wheat  @ 

ducted 
Net  profit  from  the  field 
per  acre 

"      15 

Shocking  wheat,  2  da.,  @ 

$1.75 

Aug.  30 

Threshing   480  bu.  wheat 

@,  60 

"      30 

Help  threshing,  3%  da.,  @ 

$1.50 

TOTAL 

3.  Draw  up  these  facts  into  an  account,  and  find  the  totals  and 
the  net  profit,  and  treat  in  the  same  way  as  above : 


30  RATIONAL    GRAMMAR    SCHOOL    ARITHMETIC 

Day-Book  for  Thirty- Acre  Pasture 

1.  Bought  120  bu.  corn  @  37/,  Apr.  23. 

2.  Sold  5  yearling  steers  @  §18.00,  Apr.  25. 

3.  Sold  3  yearling  heifers  @  $15.00,  July  6. 

4.  Sold  2  milk-cows  @  $46,  July  8. 

5.  Bought  4  two-yr.  old  heifers  @  $25,  July  10. 

6.  Sold  8  hogs  weighing  2064  lb.,  @  $5  per  100  lb.,  July  20. 

7.  8  head  of  hogs,  worth  $12  each,  died  July  25. 

8.  Bought  20  pigs  @  $2.50,  Aug.  10. 

9.  Sold  3  two-yr.  old  colts  @  $72,  Sept.  30. 

10.  Sold  2  draught  horses  @  $195,  Sept.  30. 

11.  Sold  9  four-yr.  old  steers  @  $68,  Oct.  25. 

12.  Bought  10  yearling  calves  @  $12,  Nov.  1. 

13.  Sold  3  Jersey  milk-cows  @  $45,  Nov.  15. 

14.  Paid  6  mo.  wages  for  one  man  @  $30,  Dec.  23. 

15.  Sold  8  head  of  hogs  @  $15,  Jan.  10. 

16.  Bought  12  head  of  hogs  @  $9.50,  Feb.  14. 


NOTATION   AND   NUMERATION 

§18.  Digits.— 0,  1,  2,  3,  4,  5,  6,  7,  8,  9. 

1.  How  many  characters  are  given  above?  By  what  names  do 
you  know  the  first  character?  We  shall  call  it  zero. 

These  ten  characters  are  called  digits  o?  figures;  and  by  means 
of  them  all  numbers,  no  matter  how  large,  are  expressed. 

§19.  Place  and  Name  Values  of  Digits. 

1.  Read  the  following  numbers: 

$30  3  30  30,000 

300  mi.  .3  300  300,000 

3  yd.  .03  3000  T\ 

"What  is  the  same  in  all  these  numbers?  What  is  different  in 
the  numbers? 

2.  Eead: 

7  feet      7  days        7  hundreds      7  millions      7  tenths 

How  many  feet?  7  what?  How  many  hundreds?  7  what? 
How  many  millions?  7  what?  In  all  these  cases,  what  is  the 
same?  What  is  different? 

3.  In  the  number  666,  what  does  the  first  6  on  the  right 
denote?  the  second  6?  the  third?         1st  Ans.  6  ones  or  6  units. 

4.  If  the  value  which  depends  on  the  name  of  a  digit  is  called 
its  -name  value,  what  shall  we  call  the  value  which  depends  on  its 
place? 

5.  How  many  values  has  a  digit?     What  gives  the  first  value? 
the   second?      Which    is   the   permanent   value?      Which    is  the 

changeable  value? 

t 

§20.  ORAL    WORK 

1.  Give  the  place  value  of  2  in  each  of  the  following  numbers : 

20         2         200         .2         20,000         2000         2222 

2.  In  286,  for  what  number  does  the  2  stand?  the  8?  the  6? 

31 


32  RATIONAL   GRAMMAR    SCHOOL   ARITHMETIC 

We  may  read  the  "number  2  hundreds,  8  tens,  6  units,  or  two 
hundred  eighty-six,  meaning  two  hundred  eighty-six  units. 
Writing  the  names  of  the  three  places,  they  appear: 


III 

286 
meaning  200  +  80+6. 

3.   Read  the  following  numbers,  first  as  hundreds,  tens,  and 
units,  and  then  as  units: 

729       916       335       494 
845       642       538       739 

§21.  Periods. — For  convenience  in  reading,  the  digits  of  large 
numbers  are  grouped  into  periods  of  three  digits  each.  These 
periods  have  names  depending  upon  their  position ;  but  the  three 
places  composing  each  period  are  always  hundreds,  tens,  and  units.* 
The  first  seven  periods  are  units,  thousands,  millions,  billions, 
trillions,  quadrillions,  and  quintillions.  The  first  three  periods 
are  sufficient  for  common  purposes. 

QUINTIL-         QUADHIL-  TRIL-  BlL-  MIL-  THOU-  TTVTT« 

LIONS  LIONS  LIONS  LIONS  LIONS  SANDS 


M  T  x 

I  1 .  *a    1  1  *a    I  1  E    I  S  E    !  1  *s    §  i  13    1 . 1  *a 

H  HP   H  H  P   s  ^  P   w  £  P   a  £  P   WHP   w  $  P 
253   462  571   324   918   645   284 

WRITTEN   WORK 

Write  the  following  in  figures : 

1.  Forty -five  thousand,  two  hundred  eighty -four;  six  thousand, 
nine  hundred  thirty-four ;  three  hundred  twenty-one  thousand,  one 
hundred  thirteen ;  five  thousand,  five. 

NOTE. — Fill  all  vacant  places  with  zeros. 

*In  Great  Britain  a  notation  different  from  that  explained  in  §21  is  used.  The 
method  of  §21  is  called  the  French  notation  and  is  used  altogether  in  the  United  States, 
France,  and  Germany.  The  English  notation  groups  the  digits  into  periods  of  six 
figures  each : 

Billions  Millions  Units 


In  the  English  notation  a  million  millions  make  a  billion,  a  million  billions  a 
trillion,  etc.  In  the  French  notation  a  thousand  millions  make  a  billion,  a  thousand 
billions  a  trillion,  etc. 


NOTATION    AND    NUMERATION  33 

2.  Nine  tnousand,  ten ;  ten  thousand,  nine ;  six  hundred  thou- 
sand, six  hundred;  five  million,  fifty  thousand,  three ;  eighty  thou- 
sand, eighty. 

§22.  Decimal  Notation.       ORAL  WORK 

1.  10  equals  how  many  units?    1  equals  what  part  of  10? 

2.  1  hundred  equals  how  many  tens?  how  many  units?    1  ten 
equals  what  part  of  1  hundred?    1  unit  equals  what  part  of  1  hun- 
dred? 

3.  1  thousand  equals  how  many  hundreds?     1  hundred  equals 
what  part  of  1  thousand? 

4.  The  number  denoted  by  each  digit  in  1111  is  how  many 
times  as  great  as  the  number  denoted  by  the  digit  to  its  right? 
The  number  denoted  by  each  digit  equals  what  part  of  the  num- 
ber denoted  by  the  digit  to  its  left? 

5.  How  many  tenths  in  1  unit?     1  unit  equals  how  many  times 
1  tenth  (.1)? 

6.  1  hundredth  (.01)  equals  what  part  of  1  tenth?     1  tenth 
equals  how  many  times  1  hundredth? 

7.  How  does  the  place  value  of  the  unit  change  from  right  to 
left?  from  left  to  right? 

§23.  Reading  Numbers. 

1.  Read  the  following  numbers : 

23,910  800,007 

20,390  1,001,001 

3,007  .6 

365,834  .27 

4,004,246  1.07 

Mean  radius  of  the  earth  636,739,510  centimeters 

Mean  distance  to  moon  76,429,120  rods 

Mean  radius  of  sun  138,560,000  rods 

Mean  distance  to  sun  92,897,500  miles 

Mean  distance  to  sun  =  149,500,000,000  meters 

Light  travels  in  one  minute  -  11,199,000  miles 


34  RATIONAL   GRAMMAR   SCHOOL    ARITHMETIC 

Number  of  vibrations  for  red  light  =   482,000,000,000  per  second 
Number  of   vibrations  for  green 

light  =   584,000,000,000  per  second 

Number  of  vibrations  for    violet 

light  =    707,000,000,000  per  second 

Mass  of  moon  =  75,000,000,000,000,000,000  tons 

Mass  of  sun  .  26,892,000  moon  masses 

Distance  from  sun  to  nearest  star  =  18,600,000,000,000  miles 

§24.  Writing  Numbers.     WRITTEN  WORK 

Write  in  numerals  the  following : 

1.  Four   billion,  two  hundred  thirteen  million,  one  hundred 
twenty-two  thousand,  six  hundred  seven;  forty-five  billion,  four 
hundred  fifty  million,  three  hundred  twenty-seven  thousand,  one. 

2.  Seven  hundred  eighteen  billion,  two  hundred  million ;  one 
hundred  ninety  thousand,  four  hundred  two ;  three  million,   one 
hundred  twenty-four;   fifteen   thousand,  nine;  ninety;  one  hun- 
dred eighty-six  billion;  one  hundred  forty-seven  million,  one  hun- 
dred thousand,  one  hundred. 

NOTE. — Persons  who  work  with  long  numbers,  containing  many 
zeros  use  a  shorter  way  of  writing  them.  They  write  25,000,000,000  thus, 
25  X  109,  and  675,000,000,000,000  is  written  675  X  1012.  The  small  9  or  12, 
written  above  and  to  the  right  of  the  10,  tells  how  many  zeros  are  to 
be  written  after  the  first  part  of  the  number. 

This  is  called  the  index  notation. 

Write  in  this  index  notation  the  numbers  above  which  contain  sev- 
eral zeros. 

The  digits,  excepting  zero,  were  first  used  by  the  Hindus,  but 
they  were  introduced  into  Europe  by  the  Arabs,  and  we  call  the 
numbers  written  with  these  digits  Arabic  numerals.  It  would  be 
more  just  to  call  them  Hindu  numerals. 

§25.  Roman  Notation. — The  Romans  used  letters  instead  of  digits 
to  represent  numbers.     This  notation  is  still  used  upon  monu- 
ments, and  in  numbering  chapters  and  volumes  of  books. 
In  the  Roman  notation, 

1=1  L  =    50  D  =    500 

V  =    5  0  =  100  M  =  1000 

X  =  10 


ADDITION 


35 


When  a  letter  of  less  value,  as  I,  is  placed  before  a  letter  of 
greater  value,  as  V,  the  value  of  the  less  is  to  be  taken  from  that 
of  the  greater;  thus,  IV  means  4  and  XC  means  90. 

When  a  letter  of  less  value  follows  one  of  greater  value,  its 
value  is  added  to  that  of  the  greater;  as,  VI  for  6,  XV  for  15, 
and  CX  for  110. 

Repeating  a  letter  repeats  its  value;  as,  XX  for  20,  CO  for  200. 

A  horizontal  line  over  a  letter  increases  its  value  a  thousand- 
fold; as,  C  meaning  100,000,  D  meaning  500,000. 

In  interpreting  numbers  written  in  the  Roman  notation,  always 
begin  on  the  right. 

§26.  Change  of  Notation.     WRITTEN  WORK 

1.  Change  from  Roman  to  Arabic: 

XCV  CMXIX  CDLXXXIV 

DXXV  MLXXX  MDCCLXXVI 

DCIV  MCCLXX  LXIX 

XCIX  DLVI  CCXI 

2.  Change  from  Arabic  to  Roman: 

10  65  100  572  619  1902 

30  77  500  78  584  1774 

60  80  140  50  400  1565 

49  95  46  246  2000  2590 


}27.  Definitions. 


ADDITION 


ORAL    WORK 


1.  A  bicyclist  rides  19  mi.  Monday  and  10  mi.  Tuesday;  how 
far  does  he  ride  both  days? 

2.  I  paid  $28  for  a  suit  of  clothes  and  $20  for  a  bicycle;  how 
much  did  I  pay  for  both? 

3.  A  four-sided  lot  has  sides  of  the  lengths:  25  yd.,  4  yd.,  11 
yd.,  and  28  yd. ;  how  far  is  it  around  the  lot? 

4.  A  man  works  for  me  7  hr.  one  day,  6  hr.  the  next  day,  7 
hr.  the  next,  and  8  hr.  the  next;  how  many  hours  does  he  work 
for  me  in  all? 


36  RATIONAL   GRAMMAR   SCHOOL   ARITHMETIC 

5.  A  newsboy  sold  18  papers  Wednesday,  12  papers  Thursday, 
and  16  papers  Friday;  how  many  papers  did  he  sell  in  all? 

6.  The  number  of  hours  the  sun  shone  on  each  of  the  days  of 
a  certain  week  was:  8  hr.,  6  hr.,  3  hr.,  7  hr.,  5  hr.,  9  hr.,  2  hr. 
How  many  hours  of  sunshine  were  there  during  the  week? 

7.  A  farmer's  fields  are  of  the  following  sizes  :    40  acres,  80 
acres,  16  acres,  and  34  acres.     How  many  acres  in  all? 

8.  How  many  dollars  are  $25  and  $35  and  $20? 

9.  How  many  feet  are  12  ft.,  8  ft.,  6  ft.,  12  ft,  and  4  feet? 

The  answer  to  problem  1  is  the  same  as  the  answer  to  the 
question:  "What  single  distance  is  just  as  long  as  the  distances 
19  mi.  and  10  mi.  combined?"  Answering  problem  2  answers  the 
question:  "What  single  amount  equals  $28  and  $20  combined?" 

Combining  numbers  into  a  single  number  is  addition. 

The  result  of  addition  is  called  the  sum  or  amount. 

The  numbers  to  be  combined  or  added  are  the  addends. 

Thus  in  the  problem 


Addend. 

29  mi.     Sum 

19  mi.  and  10  mi.  are  the  addends  and  29  mi.  is  the  sum. 

The  sign  of  addition  is  -f.  It  is  read  "plus."  In  some  cases 
it  may  be  read  "and,"  though  it  is  better  to  use  the  correct  read- 
ing, "plus,"  at  all  times. 

The  sign  =  when  written  between  two  numbers  means  that  they 
are  equal.  It  should  never  be  read  "is"  or  "are,"  but  always 
"equals"  or  "is  equal  to." 

10.  Eead  and  answer  these  questions  : 

(1)3  +  8  +  4  =  ?  (4)  12  +  7  +  8  =  ?  (7)  3  +  15  +  4  +  9  =  ? 
(2)7  +  6  +  8  =  ?  (5)  6  +  13  +  7  =  ?  (8)6  +  16  +  8  +  9  =  ? 
(3)9  +  8+5  =  ?  (6)  5  +  18  +  9  =  ?  (9)5  +  14  +  9  +  7  =  ? 

11.  In  these  questions,  what  numbers  should  stand  in  place  of 
the  question  mark? 

(1)  2  8's  +  5  8's  =  ?  8's.  (5)  9  #'s  +  8  y's  =  ?  y's. 

(2)  6  9's  +  7  9's  =  ?  9's.  (6)  15  z's  +  10  z's  =  ?  z's. 

(3)  12  13's  +  7  13's  =  ?  13's.  (7)  8  a's  +  17  a's  =  ?  fl's. 
(4)*  7  z's  +  5  «'s  =  ?  »'s.  (8)  16  i's  +  20  J'g  =  ?  J's. 

*  Read  4  thus,  "7  cc's  plus  5  cc's  equal  how  many  x's?" 


ADDITION  37 

Exercises.  WBITTEX  WORK 

1.  In  five  trips  a  street-car  carried  the  following  number  of 
passengers:     30,  42,  45,  60,  65.     Find  the  whole  number  of  pas- 
sengers. 

When  the  sums  can  not  be  seen  mentally  it  is  convenient  to 
arrange  the  addends  in  this  form: 

CONVENIENT- 
FORM 

SOLUTION. — Writing  the  addends  so  that  units,  tens, 
hundreds,  etc.,  shall  stand  in  the  same  vertical  column, 
we  add  the  units  first;  thus,  5,  10,  12  units  =  1  ten  and 
2  units.  Write  the  2  units  in  units  column  under  the  line 
and  the  1  ten  in  tens  column.  Then,  6,  12,  16,  20,  S3  tens 
=  2  hundreds  and  3  tens.  Write  the  3  in  tens  column 
and  the  2  in  hundreds  column.  Then  add  the  partial  12 

sums  as  shown.  23 

242 

2.  On  these  trips  the  conductor  collected  $1.50,  $2.10,  $2.25, 
$3.00,  $3.25.     The  fares  for  the  five  trips  amounted  to  what? 

3.  In  one  week  my  street-car  fare  was  350,  250,  400,  250, 
300,  300.    Find  the  whole  amount. 

4.  In  one  evening  a  telegraph  operator  sent  messages  contain- 
ing the  following  numbers  of  words:   2,  6,  3,  12,  6,  3,  3.     What 
was  the  total  number  of  words  dispatched? 

5.  Fred  bought  a  note  book  for  100,  a  lead  pencil  for  50,  a 
foot  rule  for  100,  a  compass  for  250,  and  a  bottle  of  ink  for  100. 
What  was  the  amount  of  his  bill? 

6.  A  train  moves  the  following  distances  in  five  successive 
hours:  40  mi.,  45  mi.,  50  mi.,  50  mi.,  48  mi.    What  distance  is 
traveled  in  the  given  time? 

Find  the  total  sales  in  each  of  the  following  problems : 

7.  Silk  sales:  2|  yd.,  3£  yd.,  f  yd.,  2J-  yd.,  3  yards. 

8.  Linen  sales:  2J  yd.,  5  yd.,  6  yd.,  2£  yd.,  5  yards. 

9.  Ribbon  sales:  1£  yd.,  \  yd.,  3  yd.,  2  yd.,  12  yards. 

10.  Thread  sales:    25  spools  white  cotton,  6  spools  black  cot- 
ton, 3  spools  black  linen,  4  spools  A  silk,  6  spools  twist. 

11.  Handkerchief  sales:    6,  12,  3,  6,  12,  6,  3,  2.     How  many 
dozen  were  sold? 


38 


RATIONAL   GRAMMAR   SCHOOL   ARITHMETIC 


12.  Notion  sales:    6  papers  pins,  2  papers  safety  pins,  6  papers 
needles,  2  cards  darning  wool,  2  combs,  12  bunches  tape,  4  bunches 
braid.     How  many  articles  were  sold? 

13.  Coffee  sales:   -J  lb.,  2  lb.,  2£-  lb.,  3  lb.,  f  lb.,  l\  lb.,  5  lb., 
4  pounds. 

14.  Sugar  sales:  18  lb.,  12  lb.,  12  lb.,  20  lb.,  5  lb.,  6  pounds. 

15.  Flour  sales:  50  lb.,  150  lb.,  2S  lb.,  15  lb.,  25  pounds. 

16.  Potato  sales:    2  bu.,  1£  bu.,  4  bu.,  6  bu.,  2  bu.,  1£  bu.,  3 
bushels. 

17.  Apple  sales:    2  bu.,  1  bu.,  3  pk.,  2  pk.,  1  pk.,  2  bu.,  1£ 
bu.  ,  4  bushels.     (Answer  in  bushels.  ) 

18.  Egg  sales:    7  doz.,  3  doz.,  6  doz.,  4  doz.,  8  doz.,  5  doz.,  2 
dozen. 

19.  Butter  sales:    5  lb.,   3  lb.,  3  lb.,   2£  lb.,l|lb.,   3  lb.,   5 
pounds. 


§29.  Problems. 

1.  A  fifth  grade   staked  out    a 
-275'—  -- *•      rectangular  plot   of   ground   to  be 

used  for  a  playground  and  school 
^  garden.     The  eighth   grade   pupils 
?  fenced  the  plot  in  and  ran  a  cross 
fence   separating   the   parts.       The 
entire  plot  was  126  ft.  wide  by  275 
ft.  long   (126'  x  275').      How  many 
feet  of  fence  were  needed? 


f 

| 

1 

C 

j 

Playground 

.7 

*i               Garden 

; 

! 

I 

1 

FIGURE  19 


SOLUTION.  —  Arrange  the  addends  in  convenient 
form.  Beginning  at  the  bottom  of  units  column, 
think  the  sums  thus,  12,  17,  23,  28  equals  2  tens 
and  S  units.  Write  the  8  in  units  column.  Add 
the  2  tens  to  the  tens  digits,  thinking  (not  speaking) 
4,  6,  13,  15,  2'2  tens,  equals  2  hundreds  and  2  tens. 
Write  the  2  tens  in  tens  place  and  add  the  hundreds 
to  the  hundreds  in  the  third  column,  thus  3,  4,  6,  1 
9  hundreds.  Write  the  9  in  hundreds  place,  and 

since  we  are  adding  feet  the  sum  is  928  ft.  Test  the  correctness  of  the 
sum  by  beginning  at  the  top  of  each  column  and  adding  downward.  If 
the  sum  is  the  same  the  work  is  probably  correct.  This  is  called  check- 
ing the  work. 

2.  Add  187,  892,  and  478. 


CONVENIENT  FORM 

275  ft.  1 

126  ft!  I 

275  ft.  }•  Addends 

126  ft.  I 

126  ft!  J 

^x  f<-      Q 


ADDITION 


39 


3.  In  a  certain  school  there  were  enrolled  one  month : 
96  children  in  the  kindergarten      100  in  the  fifth  grade 


182  in  the  first  grade 
143  in  the  second  grade 
133  in  the  third  grade 
123  in  the  fourth  grade 


1)5  in  the  sixth  grade 
83  in  the  seventh  grade 
76  in  the  eighth  grade 


How  many  pupils  were  in  the  whole  school? 
4.  During  the  same  month,  the  numbers  of  cases  of  tardiness 
were  as  follows : 


Kindergarten ...  10 

First  grade 12 

Second  grade...     4 


Third  grade 8 

Fourth  grade 5 

Fifth  grade 9 


Sixth  grade  12 

Seventh  grade..     4 
Eighth  grade ...     2 


What  was  the  total  number  of  cases  of  tardiness? 

5.  The  numbers  of  cases  of  absence  for  the  same  month  were 
as  follows,  beginning  with  the  kindergarten:  87,   60,  46,  21,  15, 
20,  15,  14,  9.     Find  the  total  number  of  cases. 

Teachers  may  take  reports  of  their  own  or  of  other  schools  and 
make  similar  problems. 

6.  Fill  out  for  each  of  these  cities  the  total  number  of  rains 
during  May  and  June,  1902,  and  the  total  rainfall  in  inches.     Do 
not  rewrite  the  numbers. 


CITY 

NUMBER  or  RAINS 

RAINFALL  IN  INCHES 

MAY 

JUNE 

TOTAL 

MAY 

JUNE 

TOTAL 

Chicago  

11 
13 
14 
13 
13 
10 

15 
16 
18 
15 
16 
13 

4.46 
4.57 
3.46 
5.02 
2.83 
2.21 

6.00 
6.76 
6.16 
4.19 
7.29 
9.08 

Des  Moines 

Detroit 

Kansas  City  

Omaha  

St.  Louis  

The  length  in  feet,  the  number  of  officers  and  men,  the  dis- 
placement in  tons,  and  the  indicated  horse  power  of  eight  of  the 
first-class  battleships  of  the  United  States  navy  are  as  given  in  the 
table  on  the  following  page. 


40 


RATIONAL    GRAMMAR    SCHOOL   ARITHMETIC 


NAME 

LENGTH 

MEN 

DISPLACE- 
MENT 

HORSE 
POWER 

Alabama  

368 

585 

11  525 

11  366 

Wisconsin 

368 

585 

11  525 

10  000 

Kearsarge 

368 

520 

11  525 

11  954 

Kentucky 

368 

520 

11  525 

12*318 

Iowa  

360 

444 

11  340 

12  105 

Indiana  
Massachusetts  

348 
348 

465 
424 

10,288 
10  288 

9,738 
10  403 

Oregon 

348 

424 

10  288 

11  111 

Total  

7.  If  these  8  vessels  stood  in  a  straight  line  with  their  ends 
touching,  how  far  would  they  reach? 

8.  How   many   officers   and   men  are   needed  to   man   the  8 
ships? 

9.  The   displacement   is   the   number   of  tons  of    water   the 
vessel  pushes  aside  as  it  floats.     What  is  the  combined  displace- 
ment? 

10.  What  is  the  combined  horse  power  of  the  engines  of  the  & 
ships? 

11.  At  the  close  of  1900  the  numbers  of  teachers,  schools,  and 
pupils  in  the  6  provinces  of  Cuba  were  as  follows : 


PROVINCE 

TEACHERS 

SCHOOLS 

PUPILS 

Havana  

904 

904 

41  383 

Puerta  Principe  

246 

246 

9,355 

Santa  Clara 

879 

879 

44  872 

Santiago 

645 

645 

33  983 

Pinar  del  Rio.                       .  .            .... 

274 

274 

13  282 

Matanzas.   ...            .  .           

619 

619 

29  398 

Total 

Find  the  totals  and  tell  what  they  mean. 

12.  The  6  European  countries  from  which  most  immigrants 
came  to  the  United  States  in  1901  were  as  follows : 


ADDITION 


41 


COUNTRY 

IMMIGRANTS  IN  1901 

IMMIGRANTS  IN  1900 

MALE 

FEMALE 

TOTAL 

MALE 

FEMALE 

TOTAL 

Italy           

106,306 

78,725 
54,070 
12,894 

12,875 

29,690 
34,665 
31,187 
17,667 
10,456 

76,088 
79,978 
60,091 
16,672 
10,262 

24,047 
34,499 
31,066 
19,058 

8,388 

Austria-Hungary  
Russia                      •    •  • 

Ireland      

Sweden  

Total  

. 

Fill  out  all  the  totals  and  tell  what  they  mean. 
13.  The  published  daily  circulation  of  a  city  newspaper  from 
week  to  week  is  here  given : 


FIRST  WEEK 

SECOND  WEEK 

THIRD  WEEK 

FOURTH  WEEK 

FIFTH  WEEK 

DAY   COPIES 

1...  255,  572 
"2...  303,  062 
3...  323,513 
4...  312,724 
5...  306,  009 
6...  297,  918 

DAY   COPIES 

8...  305,  725 
9...  303,  489 
10...  304,  991 
11.,.  304,  746 
12...  305,  515 
13...  299,  095 

DAY  COPIES 

15...  304,  636 
16...  302,  173 
17...  302,  650 
18...  304,  255 
19...  302,  942 
20...  297,  684 

DAY  COPIES 

22...  304,  836 
23...  304,  698 
24...  310,  870 
25...  299,  780 
26...  301,455 
27...  298,  446 

DAY    COPIES 

29...  303,  383 
30...  302,  005 

Total 

Total 

Total 

Total 

Total 

Total 
for  mo. 

Find  the  weekly  totals  and  the  total  for  the  whole  month. 

14.  In  the  year  1891  the  United  States  imported  94,628,119  Ib. 
of  coffee,  which  was  54,262,757  Ib.  less  than  was  imported  during 
the  two  previous  years.     How  many  pounds  were  imported  during 
the  two  previous  years? 

15.  The  United  States  imported  79,192,253  Ib.  of  tea  in  1889; 
83,494,956  Ib.  in  1890;  and  82,395,924  Ib.  in  1891.     How  many 
pounds  of  tea  were  imported  in  the  three  years? 

16.  In  1889  $27,024,551  worth  of  molasses  was  imported  into 
the    United  States;   in   1890  $31,497,243    worth;    and  in   1891 
$2,659,172  worth.     How  many  dollars'  worth  was  imported  during 
the  three  years? 


RATIONAL    GRAMMAR   SCHOOL   ARITHMETIC 


17.  Of  the  yearly  internal  trade  of  New  York  state,  $1,050,- 
000,000  worth  of  the  freight  passes  over  the  railroads;  $150,000,- 
000  over  the  eauals;  and  $250, 000, 000  over  Long  Island  Sound 
and  the  lakes.  What  is  the  total  value  of  the  internal  trade? 


THE  TEN  LARGEST  CITIES  OF  THE 
UNITED  STATES 

POPULATION 

New  York 3,437,202 

Chicago 1,698,575 

Philadelphia 1,293,697 

St.  Louis 575,236 

Boston' 560,892 

Baltimore 508,957 

Cleveland 381,768 

Buffalo 352,387 

San  Francisco 342,782 

Cincinnati 325,902 


THE  TEN  LARGEST  FOREIGN  CITIES 
POPULATION 

London 4,433,018 

Paris   2.536,834 

Canton   2,000,000 

Berlin 1,677,304 

Vienna  1,364,548 

Tokyo 1,268,930 

St.  Petersburg 1,267,023 

Pekin 1,000,000 

Moscow 988,610 

Constantinople 900,000 


18.  Find  the  total  population  of  the  ten  largest  cities  of  the 
United  States. 

Test  the  correctness  of  your  addition  hy  adding  the  columns 
from  top  downward. 

Teachers  may  omit  the  following  method  of  checking  if  thought  too 
difficult. 

A  very  useful  method  of  checking  long  problems  in  addition  is  known 
as  casting  out  the  nines. 

To  cast  out  the  nines  from  a  number  add  its  digits  and  whenever  the 
sum  equals  or  exceeds  9,  drop  9  and  continue  adding  the  next  digits  to 
what  remains,  dropping  9  whenever  the  sum  equals  or  exceeds  9.  The 
last  remainder  is  called  the  excess. 

ILLUSTRATION.— Cast  the  9's  out  of  647,255. 

Beginning  on  the  left,  6  +  4=10;  drop  9,  giving  the  remainder  1. 
1  +-  7  +  2  =  10,  drop  9.  1  +  5+5=11,  drop  9.  The  last  remainder  is  2 
and  this  is  the  excess  of  647,255. 

To  check  addition  by  casting  out  nines,  cast  the  nines  out  of  the 
addends  and  the  sum.  Then  cast  out  the  nines  from  the  excesses  of  the 
addends.  If  this  last  excess  of  the  excesses  equals  the  excess  of  the  sum, 
the  work  is  probably  correct. 


ILLUSTRATION 
8,465 
3,282 
4,497 
2,957 
7,642 


EXCESS 
5 
6 
6 
5 
1 


26,843  5  —  5 

5  =  excess  of  the  sum. 

5  =  excess  of  ^xcesses  of  addends. 


To  obtain  the  excess  of  the 
excesses  of  the  addends : 

5  +  6  =  11,  drop  9,  giving  2. 
2  +  6  +  5  =  13,  drop  9,  giving  4. 
4  +  1  =  5,  the  excess  of  the  ex- 
cesses of  the  addends.  Since  this 
equals  the  excess  of  the  sum,  the 
work  is  checked. 


ADDITION 


43 


10.  Find  the  population  of  the  ten  largest  cities  of  Europe  and 
Asai.     (See  table,  p.  42.) 

20.  Find  the  total  population  of  these  twenty  cities. 

21.  Find  the  population  of  those  cities  in  both  lists  having 
more  than  900,000  inhabitants. 

22.  Make  and  solve  similar  problems  based  on  the  table. 

§30.  Distribution  of  Population  in  the  United  States  in  1900. 


FIGURE  20 

On  the  map  the  main  geographical  divisions  are  surrounded  by 
heavy  full  lines :  — .  The  divisions  are  separated  into  sections 

by  heavy  dotted  lines : .  Thus  the  New  England  states  are 

the  eastern  section  of  the  North  Atlantic  division;  and  New  York, 
Pennsylvania,  and  New  Jersey  are  the  western. 

Teachers  may  select  such  problems  from  this  list  as  have  a 
special  geographical  interest  to  their  school. 

1.  From  the  table  of  statistics  given  on  the  following  page, 
find  the  total  population  of  the  New  England  states  for  1900. 

2.  Of  the  North  Central  division. 

3.  Of  the  Western  division. 

4.  Of  the  United  States  with  outlying  territory. 


KATIONAL   GRAMMAR    SCHOOL   ARITHMETIC 


5.  Find  the  total  area  in  square  miles  of  the  United  States 
with  outlying  territory. 

AREA  AND  POPULATION  OP  STATES  AND  TERRITORIES 


STATE  OB 
TERRITORY 

AREA 
IN  SQ. 
MILES 

POPULA- 
TION 
1900 

PUPILS  IN 
ELEM.  & 
SEC. 
SCHOOLS 

STATE  OR 
TERRITORY 

AREA 
IN  SQ. 
MILES 

POPULA- 
TION 
1900 

PUPILS  IN 
ELEM.  & 
SEC. 
SCHOOLS 

Me.  . 
N.  H  
Vt 

33,040 
9,305 
9,565 
8,315 
1,250 
4,990 
49,170 
7,815 
45,215 

694,666 
411,588 
343,641 
2,805,346 
428,556 
908,420 
7,268,894 
1,883,669 
6,302,115 

130,918 
65,193 
65,964 
474,891 
64,537 
155,228 
1,209,574 
315,055 
1,151,880 

Ky  
Tenn.    .  .  . 
Ala 

40,400 
42,050 
52,250 
46,810 
48,720 
265,780 
39,030 
53,850 
31,400 

2,147,174 
2,020,616 
1,828,607 
1,551,270 
1,381,625 
3,048,710 
398,331 
1,311,564 
392,060 

501,893 
485,354 
376,423 
360,177 
196,169 
578,418 

314,662 
99,602 

Mass  
R  I 

Miss  
La  
Tex  
Okl  
Ark  

Conn  

N.  Y  
N  J 

Penn  

Ind.  T.  .  .  . 

North    At- 
lantic divi- 
sion 

South  Cen- 
tral    divi- 
sion 

Del  .. 

2,050 
12,210 
70 
42,450 
24,780 
52,250 
30,570 
59,475 
58,680. 

184,735 
1,188,044 
278,718 
1,854,184 
958,800 
1,893,810 
1,340,316 
2,216,331 
528,542 

33,174 
229,332 
46,519 

358,825 
232,343 
400,452 
281,891 
482,673 
108,874 

Mont.    .  .. 
Wy  
Col  
N.  M  

146,080 
97,890 
103,925 
122,580 
113,020 
84,970 
110,700 
84,800 
69,180 
96,030 
158,360 

243,329 
92,531 
539,700 
195,310 
122,931 
276,749 
42,335 
161,772 
518,103 
413,536 
1,485,053 

39,430 
14,512 
117,555 
36,735 
16,504 
73,042 
6,676 
36,669 
97,916 
89,405 
269,736 

Md  
D.  C  

Va 

W.Va.... 
N.  C  

so.  .  . 

Ariz  

Utah 

Nev  
Idaho  
Wash.  ... 
Ore    

Ga. 

Fla  

South    At- 
lantic divi- 
sion   

Cal  

Western 
division  — 

Ohio  
Ind  
Ill  
Mich  
Wis  
Minn  .  ... 
Iowa  
Mo 

41,060 
36,350 
56,650 
58,915 
56,040 
83,365 
56,025 
69,415 
70,795 
77,650 
77,510 
82,080 

4,157,545 
2,516,462 
4,821,550 
2,420,982 
2,069,042 
1,751,394 
2,231,853 
3,106,665 
319,146 
401,570 
1,066,300 
1,470,495 

829,160 
564,807 
958,911 
498,665 
445,142 
399,207 
554,992 
719,817 
77,686 
96,822 
288,227 
389,583 

U.  S.  with- 
out outlying 
territory  — 

Alaska  ... 
Hawaii... 
Phil.  Is.. 
Tutuila  .  . 
Guam  .  .  . 
Porto      ) 
Rico  f 

590,884 
6,449 
114,410 

77 
150 

3,531 

63,592 
154,001 
6,961,339 
6,100 
9,000 

953,243 

N.D  
S.  D  
Neb  
Kan  

North  Cen- 
tral divi- 
sion   

Total   U.  S. 
with    outly- 
ing territory 

ADDITION  45 

6.  Find  the  area  in  square  miles  of  the  North  Central  division. 

7.  Of  the  South   Central  division.      Which  has  the  greater 
territory? 

8.  What  state  in  the  Union  contains  the  smallest  number  of 
square   miles?   the    greatest?      What   state    supports    (contains) 
the  greatest   population?    the   smallest?     How  do  the   areas  of 
these  two  states  compare? 

9.  Find  the  number  of  pupils  attending  school  in  the  New 
England  states;  in  the  North  Central  division. 

10.  Find  the  total  number  of  children  attending  school  in  the 
United  States. 

11.  Make  such  original   problems  as   these:    Find  the  total 
population  in  1900  of  the  13  original  states;  the  total  area  of  the 
13    original   states;  of  the   states   east   of  the  Mississippi  river, 
etc. 

12.  Check  the  correctness  of  your  work  in  problems  2  and  7 
by  adding  the  columns  first  as  a  whole  and  then  adding  the  foot- 
ings of  the  separate  sections  and  comparing  the  two  sums. 

13.  Find  the  population  and  the  area  of  the  eastern  North 
Atlantic  states ;  of  the  western. 

14.  Find  the  number  of  pupils  attending  the  elementary  and 
secondary  schools  in  both  the  sections  mentioned  in  problem  13. 

15.  Check  the  work  of  problems  1  and  4  by  casting  out  the  nines. 

§31.  Measurements. 

A  square  unit,  or  a  unit  square,  is  a  square  each 
of  whose  sides  is  1  unit  long. 

The  area  of  a  surface  is  the  number  of  square 
units  in  it. 

1.  What  is    a  square  inch?    a   square   foot?    a 

square  yard?  a  square  mile?  a  square 
rod? 

2.  If  a  rectangle  is  made  up  of  5 
rows  of  10  unit  squares  each,  what 
is  its  area?  How  many  tens  of  unit 


FIGURE  22  squares  are  in  the  rectangle? 


4C  RATIONAL   GRAMMAR   SCHOOL   ARITHMETIC 


3.  If  a  rectangle  is  12  units  long  and 
6  units  wide,  how   many  twelves  of  unit 
squares  are  there  in  the  area? 

4.  If  a  rectangle 

is  x  units  long  and 
FIGURE  23 

9    units   wide,    how 
many  z's  of  unit  squares  are  in  its  area? 


NOTE. — Seven  x's  is  written  1x  and  is    read  FIGURE  24 

"seven  a?." 

5.  How  wide  must  a  rectangle  z  inches  long  be  to  contain  6z 
sq.  in.?   152  square  inches? 

6.  How  long  must  a  rectangle  15  in.  wide  be  to  contain  I5a 
square  inches? 

7.  The  area  of  one  rectangle  is  Sz  sq.  in.  and  of  another  7z  sq. 
in. ;  how  many  square  inches  in  their  sum? 

8.  How  many  9's  are  8  9's  and  7  9's?     How  many  12's  are 
8  12's  and  7  12's?     How  many  z's  are  Sx  and  7x?  8z  +  7z  =  ? 

9.  Add  these  numbers : 

(1)  9a        (2)   Sa        (3)  26z       (4)  48*       (5)   76y        (6)  695 
Sa    .          60  47z  982  49z/  795 

(7)  Qm       (8)  68»     (9)  84c     QO)  73^     (11)   78G^    (12)  960s 
8w  75^  76c  39d  31 9a; 

9m 


SUBTRACTION 
§32.  Definitions.  ORAL  WORK 

1.  A  man  earns  $180  a  month  and  spends  $140.     What  is  the 
difference  between  his  earnings  and  expenditures? 

2.  One  farmer  owns  320  A.  of  land,  another  80  A.     Find  the 
difference  in  size  of  the  two  farms? 

3.  From  a  cheese  weighing  43  lb.,  23  Ib.  were  sold;  how  many 
pounds  remained? 

4.  From  a  bin  containing  72  bu.  of  oats,  I  use  32  bu.     How 
many  bushejs  remain? 


SUBTRACTION  47 

5.  The  sum  of  two  numbers  is  48,  and  one  of  the  numbers  is 
28 ;  what  is  the  other? 

G.  What  is  required  in  the  first  two  problems  of  this  section? 
in  the  second  two?  in  the  fifth  problem? 

7.  What  is  subtraction?1    What  is  the  result  in  subtraction 
called? 

Subtraction  means  either  of  two  things : 

(1)  The  way  of  finding  the  difference,  or  remainder,  of   two 
numbers. 

(2)  The  way  of  finding  either  one  of  two  addends  when  their 
sum  and  the  other  addend  are  known. 

With  the  first  meaning,  the  number  from  which  we  subtract  is 
the  minuend.  The  number  to  be  subtracted  is  the  subtrahend. 
The  result  is  called  the  difference,  or  remainder. 

With  the  second  meaning,  the  known  sum  is  the  minuend. 
The  known  addend  is  the  subtrahend,  and  the  unknown  addend  is 
the  difference,  or  remainder. 

The  sign  of  subtraction  —  is  read  "minus,"  and  when  placed 
between  two  numbers  means  that  their  difference  is  to  be  found. 

The  minuend  is  always  written  before  the  sign. 

8.  Tell  what  number  the  letter  stands  for  in  these  problems : 
(!)     8-5  =  3  (:3)  16-7  =  0  (5)  15-8  =  * 

(3)  17  -9  =  3  (4)  25  -  y  =  8  (6)  16  -  z  =  8 

9.  What  digit  should  take  the  place  of  the  question  mark  in 
these  problems? 

(1)  7  S's-5  8's  =  ?8's  (4)     9  a-  G   x  =  1x 

(2)  8  12's  -  3  12's  =  ?  12's  (5)  15  x  -  7  x  =  ?  x 

(3)  4  x  -  2  x  =  ?  x  (6) |15  x  -  ?  x  =  8  x 

WRITTEX   WORK 

1.  A  traveler  has  a  journey  of  437  mi.  to  make  and  he  has 
already  traveled  199  mi.  of  it;  how  much  farther  must  he  travel? 

SOLUTION.— 437  means  400  +  30  +  7,  and  199  means  100  +90  +  9. 
We  arrange  the  numbers  conveniently,  thus : 

Minuend  437  mi. 
Subtrahend  199  mi. 

Remainder  238  mi. 

Beginning  on  the  right,  9  units  can  not  be  taken  from  7  units.  We 
take  one  of  the  3  tens  and  add  it  to  the  7  units,  giving  17  units.  Then  17 
units  —  9  units  =  8  units.  Write  8  in  units  column.  Passing  to  tans 


48  RATIONAL   GRAMMAR   SCHOOL   ARITHMETIC 

column,  we  can  not  take  9  tens  from  the  2  tens  remaining,  so  we  take  one 
of  the  4  hundreds  and  add  it  to  2  tens,  making  12  tens.  Then  12  tens  —  9 
tens  =  3  tens.  Write  3  in  tens  column.  Finally,  3  hundreds  —  1  hundred 
=  2  hundreds.  Write  the  2  in  the  hundreds  column.  The  remainder 
or  difference  is  238  miles. 

2.  How  can  you  prove  238  to  be  the  correct  remainder  or 
difference  in  problem  1? 

3.  The  minuend  is  1284,  and  the  subtrahend  is  876.     What  is 
the  difference,  or  remainder?     Prove. 

4.  When  the  water  was  dried  out  of  38  oz.  of  natural  soil,  the 
dry  soil  weighed  29  oz.     Make  a  problem  from  these  facts,  solve 
it,  and  prove  your  solution  correct. 

5.  A  subtrahend  is  99,   and  the  difference  is   338.     What  is 
the  minuend?     How  is  it  found? 

6.  From  72  hr.  subtract  1  da.  and  4  hours. 

NOTE. — A  correct  answer  to  this  problem  would  be  72  hr. — 1  da.  and  4 
hr.  But  the  answer  is  to  be  expressed  in  a  single  unit.  To  do  this,  first 
change  1  da.  and  4  hr.  to  hours.  1  da.  =  24  hr. ;  24  hr.  -f-  4  hr.  =  28  hr. 
72  hr.  —  28  hr.  =  44  hours. 

7.  From  36  ft.  take  5  yd.     What  must  first  be  done  to  obtain 
the  result  in  a  single  unit?     Make  complete  statements. 

8.  20  qt.  -  2  pk.  =  x  qt.     What  is  the  first  thing  to  be  done? 
Make  complete  statements.     Find  the  value  of  x. 

9.  From  f  yd.  take  f  yd.    What  is  the  unit  of  f  yd.?  of  f  yd.? 
What  then  must  be  done  before  subtracting?     Why? 

NOTE. — Only  numbers  expressed  in  the  same  unit  can  be  subtracted 
if  the  result  is  to  be  expressed  in  a  single  unit. 

10.  Restate  the  last  problem,  using  the  sign  of  subtraction. 

§33.  Exercises. 

1.  From  a  barrel  of  vinegar  containing  31 J  gal.,  suppose  the 
following  quantities  to  be  removed,  and  give  successive  remainders : 
6  gal. ;  2i  gal. ;  5  gal. ;  3  gal. ;  2*  gal. ;  2J  gallons. 

2.  From  a  wagon  box  containing  42  bu.  of  potatoes,  a  farmer 
sold  the  following  quantities:    2  bu.,  4  bu.,  3  bu.,  2  bu.,  5  bu., 
6  bu.,  9  bu.,  4  bu.     How  many  bushels  remained? 

3.  On  March  31  the  gas  meter  read  30,000  cu.  ft.,  and  on 
April  30,  45,000  cu.  ft.     How  many  cubic  feet  of  gas  had  been 
used  during  the  month? 


SUBTRACTION  49 

4.  A  horse  and  his  rider  weigh  1375  Ib.     The  man  weighs 
160  Ib.     What  is  the  weight  of  the  horse? 

5.  Find  and  read  the  differences  in  these  exercises: 

(1)  W  (3) 

$9274.75  $4246.28  1,024,637  yd. 

1846.19  1571.16  725,448   " 

6.  Find  the  differences : 

(1)  (2) 

6,420,014  rd.  $3472.06 

1,382,741   "  2189.24 

7.  During  one  week  a  merchant's  transactions  at  his  bank  were 
as    follows:      deposits,    $217.40,     $343.27,      $290.00,      $365.18, 
$380.24,    $415.17;    withdrawals,  $200,    $320.25,  $195.75,    $286, 
$240.15,  $395.25.     Which  item  was  the  greater  at  the  close  of  the 
week,  and  how  much? 

8.  During  one  week  of  November,  1901,  63,188  cattle  were 
received  at  the  Chicago  stock  yards;   during  the  corresponding 
week  of  1899,  32,722  were  received.     What  was  the  increase? 

9.  Great  Salt  Lake  is  4200  ft.  above  sea  level,  and  Lake  Su- 
perior 602  ft.    x  is  the  difference  in  elevation.    Find  value  of  x. 

10.  Cuba  contains  45,884  sq.  mi.,  and  the  Philippines,  114,410 
sq.  mi.    Find  the  difference  in  areas. 

For  the  numbers  for  problems  11-17,  see  the  tables,  p.  50. 

11.  Find  the   total  receipts   at  Chicago  for   the  week  Nov. 
25-30,  1901,  of  cattle;  of  calves;  of  hogs;  of  sheep. 

12.  Find  the  total  shipments. 

13.  Find  the  difference  between  receipts  and  shipments. 

14.  Compare   the  receipts  with  those  of  the  previous  week; 
with  those  of  the  corresponding  week  of  1900;  of  1899. 

15.  Compare  shipments  in  the  same  manner. 

16.  Find  total  receipts  for  one  week  in  the  four  markets,  Chi- 
cago, Kansas  City,  Omaha,  and  St.  Louis,  of  cattle;  of  hogs;  of 

ep. 

17.  Compare  Chicago  receipts  with  those  of  the  other  cities. 


50 


RATIONAL   GRAMMAR    SCHOOL   ARITHMETIC 


A  WEEK'S  RECEIPTS  AND  SHIPMENTS  OF  LIVE  STOCK. 
CHICAGO,  Nov.  30,   1901 


RECEIPTS 

CATTLE 

CALVES 

HOGS 

SHEEP 

Monday          Nov.  25  

16,078 
6,941 
17,583 

5,857 
800 

537 
735 

481 

162 
40 

39,630 
41,654 
51,275 

44,090 
25,000 

26,170 
18,297 
12,393 

12,462 
2,000 

Tuesday           '       26  

Wednesday,     *       27  

Thursday,         '       28  (Holiday)  .  . 
Friday,                    29.....  
Saturday,                 30  

Total         

63,188 
53,965 
32,722 

3,814 
1,822 
1,335 

273,426 
191,104 
159,724 

104,528 
55,046 

61,718 

Previous  week 

Corresponding  week  1900  .  . 

Corresponding  week  1899  

SHIPMENTS 

CATTLE 

CALVES 

HOGS 

SHEEP 

Monday          Nov  25 

2,578 
1,904 
5,647 

1,875 
400 

o 
1 

16 

205 
30 

7,543 
1,495 
5,712 

4,308 
5,000 

2,852 
4,518 
4,501 

2,127 
1,000 

Tuesday            4      20 

Wednesday,      *      27 

Thursday,           '      28  (Holidav)  .  . 
Friday                k      29     

Saturday            '      30 

Total 

18,699 
18,438 
9,556 

324 
352 
210 

29,486 
23,575 
16,637 

24,429 
14,332 

3,182 

Previous  week 

Corresponding  week  1900  . 

Corresponding  week  1899  . 

LIVE  STOCK  RECEIPTS  FOR  ONE  WEEK  AT  FOUR  MARKETS 


CATTLE 

HOGS 

SHEEP 

Chicago      

47,300 

201,600 

71,300 

Kansas  City  

29,300 

89,000 

14,900 

Omaha 

16  800 

65  200 

12  800 

St    Louis 

10,700 

38,100 

5  800 

Total   : 

Previous  week  

144,600 

481,700 

173,600 

Corresponding  week  1900*  

111,500 

339,800 

85,000 

Corresponding  week  1899 

91  700 

206  100 

81  300 

Corresponding  week  1898               ... 

128  400 

365,500 

95  600 

Corresponding  week  1897  

143,000 

374,100 

105,600 

SUBTRACTION  51 

18.   Compare    the    total    receipts    of    the   week   with   those 
of  the  previous  week;    of  the  corresponding  week  of  1900;  1899; 

1898;  1897. 

§34.  Geography. 

1.  Lake  Titicaca  is  12,645  ft.  above  the  level  of  the  sea,  and 
Lake  Superior  602  ft.     Find  the  difference  in  eleyation. 

2.  Mt.  Everest  is  29,002  ft.   above  sea  level,  and  Mt.  Blanc 
15,744  ft.     What  is  the  difference  in  altitude? 

3.  The  Mississippi  basin  has   an  area  of  1,250,000  sq.   mi.; 
the  Amazon,  of  2,500,000  sq.  mi.     How  much  greater  is  the  basin 
of  the  Amazon  than  that  of  the  Mississippi? 

4.  The  area  of  the  Philippines  is  114,410  sq.   mi. ;   that  of 
California,  158,360  sq.  mi.     The  state  is  how  much  larger  than 
the  islands? 

5.  Compare  the  same  islands  with   Texas;  with  Illinois;  with 
Rhode  Island.     (See  table,  p.  44.) 

6.  Of  a    population  of  6,302,115  in  Pennsylvania,  1,151,880 
are  pupils  in  elementary  and  secondary  schools.     How  many  per- 
sons are  not  attending  those  schools? 

7.  In  a  certain  year  the  Hawaiian  Islands  produced  221,694 
T.  of  sugar,  while  Cuba  produced  880,372  T.     How  much  more 
did  Cuba  produce  than  the  Hawaiian  Islands? 

8.  In    1890,   25,403    manufacturing  establishments    in    New 
York  City  paid  for  wages  $230,102,167;  for  materials,  $366,422,- 
722;   for  miscellaneous  expenses,  $59,991,710.     The  value  of  the 
products  was  $777,222,721.    ^What  was  the  total  profit  for  all  the 
establishments? 

9.  The   battle  of    Lexington    was  fought  in  1775,   and  the 
battle  of  Santiago  in  1898.     How  many  years  elapsed  between  the 
two  battles? 

For  the  numbers  for  problems  10-19,  see  table,  p.  44. 

10.  What  is  the  difference  of  the  areas  of  the  largest  and  the 
smallest  states? 

11.  How  many  more  persons  are  in  the  state  with  the  largest 
population  than  in  the  one  with  the  smallest? 


52  RATIONAL    GRAMMAR    SCHOOL   ARITHMETIC 

12.  How  many  more  children  attend  school  in  one  of  these 
two  states  than  in  the  other? 

13.  Make  and   solve   other   problems,    comparing  the    areas, 
total  populations,  and  school  populations  of  the  different  states. 

14.  Which  is  the  larger,  and  by  how  much,  the  North  Atlantic 
division  or  the  South  Atlantic  division?  the  North  Central  or  the 
South  Central?  the  South  Central  or  the  Western? 

15.  Compare  the  total  populations  of  these  groups. 

16.  How  many  persons  are  not  attending  school  in  Maine?  in 
Mississippi?     How  do  these  two  results  compare? 

17.  Find  how  many  persons  are  not  attending  school  in  your 
own  state;  in  bordering  states. 

18.  How  many  more  persons  are  not  attending  school  in  New 
York  than  in  Illinois?     How  does  this  result  compare  with  the 
difference  in  the  total  populations  of  these  states? 

19.  Make  and  solve  other  problems  of  interest  to  your  school. 

20.  Find,  by  comparing  the  populations  for  1890  (p.  53)  with 
those  for  1900  (p.  44),  the  increase  of  population  for  the  ten  years 
in  each  state  of  the  North  Atlantic  division ;  in  the  whole  division. 

21.  Find  the  increase  of  population  in  your  own  state  from  1890 
to  1900.     What  is  the  average  yearly  increase? 

22.  How  many  years  since  Illinois  was  admitted  as  a  state? 
since  your  native  state  was  admitted  into  the  Union? 

23.  Make  similar  problems  from  the  table  opposite  and  the 
table  on  p.  44,  solve  your  problems,  and  check  your  work. 

24.  Each  principal  division  of  states  is  separated  into  two  or 
more  sections  by  heavy  dotted  lines  on  the  map,  p.  43.     Find 
how  much  greater,  or  less,  the  grow'th  of  population  since  1890 
has  been  in  one  section  of  each  division  than  in  another. 

§35.  Commerce. 

1.  Find  the  total  export  trade  of  the  United  States  with  the 
twelve  countries  given  in  the  table  on  p.  54,  in  1891  and  in  1901. 

2.  Find  the  increase,  or  decrease,  for  each  country,  and  place 
the  difference  in  the  last  column.     Whenever  the  difference  is  an 
increase  write  +  before  it.     When  it  is  a  decrease  write  —  before 
it.     Make  the  subtractions  without  rewriting  the  numbers. 


SUBTRACTION 


53 


DATES  OF  ADMISSION  AND  POPULATIONS  FOR  1890  OF  STATES  AND 
TERRITORIES  GEOGRAPHICALLY  GROUPED 


DATE  OF 
ADMISSION 

POPULA- 

TION  1890 

DATE  OF 
ADMISSION 

POPULA- 
TION 1890 

Me 

Mar    15,  1820 

661  086 

Ky 

June    1   1792 

1  848  635 

N  H 

June  21,  1788 

376,530 

Tenn.  .  .  . 

June    1   1796 

1,767,518 

Vt 

Mar      4,  1791 

332,422 

Ala  

Dec.   14,  1819 

1,513,017 

Mass  .... 

Feb.     7,  1788 

2,238,043 

Miss  

Dec.   10,  1817 

1,289,600 

R  I 

May   20  1790 

345  506 

La 

April  30  1812 

1  118  587 

Conn.  .  .  . 
N  Y 

Jan.      9,  1788 
July  26   1788 

746,258 
5  997,853 

Tex  
Okl  

Dec.   29,  1845 
May     2   1890 

2,235,523 
61  834 

N  J 

Dec.   18   1787 

1,444,933 

Ark  

June  15,  1836 

1,128  179 

Penn.  ... 

Dec.   12,  1787 

5,258,014 

Ind.  T.  .  . 

June  30,  1834 

180,182 

North  , 
sion 

Atlantic   divi- 

South 
sion  . 

Central    divi- 

Del 

Dec.     7  1787 

168  493 

Mont 

Nov      8  1889 

132,159 

Md. 

April  28  1788 

1,042,390 

Wy 

July  10  1890 

60,705 

DC.. 

Mar.   30,  1791 

230,392 

Col.  .  .  . 

Aug     1    1876 

412,198 

Va  

June  25,  1788 

1,655,980 

N.  M  

Sept.    9,  1850 

153,593 

W.  Va.  .  . 

N.  C  

s  c.    . 

June  20,  1863 
Nov.  21,  1789 
Mar.   23   1788 

762,794 
1,617,947 
1,151  149 

Ariz  
Utah  

Nev 

Feb.   24,  1863 
Jan.     4,  1896 
Oct     31   1864 

59,620 
207,905 
45,761 

Ga  
Fla  

Jan.     2,  1788 
Mar.     3,  1845 

1,837,353 
391,422 

Idaho.  .  .  . 
Wash.  .  .  . 
Ore  

July     3,  1890 
Nov.  11,  1889 
Feb.   14,  1859 

84,385 
349,390 
313,767 

South   j 

Atlantic   divi- 

Cal  

Sept.    9,  1850 

1,208,130 

sion 

Westen 

i  division  

Ohio  
Ind  
111.  
Mich..  .. 

Feb.    19,  1803 
Dec.   11,  1816 
Dec.     3,  1818 
Jan.    16,  1837 

3,672,316 
2,192,404 
3,826,351 
2,093,880 

U.   S. 
lying 

without    out 
territory  

Wis  
Minn.  .  .  . 
Iowa  
Mo  .... 

May   29,  1848 
May   11,  1858 
Dec,  28,  1846 
Aug  10   1821 

1,686,880 
1,301,826 
1,911,896 
2  679  184 

Alaska  .  . 
Hawaii.  . 
Phil  Is 

July  27,  1868 
April  30,  1900 

32,052 
89,990 

N.  D  

Nov     2   1889 

182  719 

Tutuila 

S.  D  
Neb  
Kan  

Nov.    2,  1889 
Mar.     1,  1867 
Jan.    29,  1861 

328,808 
1,058,910 
1,427,096 

Guam  .  .  . 
Porto      I 
Rico  J 

North 
sion  . 

Central     divi- 

Total  U 
Iviner 

.  S.  with  out- 
territory  .  .  . 

RATIONAL   GRAMMAR    SCHOOL    ARITHMETIC 


UNITED  STATES  EXPORT  TRADE 


COUNTRY 

1901 

1891 

TEN-YEAR 
DIFFERENCE 

United  Kingdom  

$598  766  799 

$482  295  796 

Germany  

184,678,723 

90  326  332 

Canada  

107,496  522 

41,686,882 

Netherlands  

85,643,804 

31,261,766 

Mexico 

36  771  568 

15  371  370 

Italy  .  . 

34  046  201 

14  447  004 

British  Australasia  . 

30  569  814 

13  564  931 

British  Africa  

24,994  766 

3  511  668 

Japan  

21  162  477 

3  839  384 

Brazil  
Argentina 

11,136,101 
11  117  521 

15,064,346 
1  909  788 

Russia 

6  504  867 

5  400  357 

Total  

VALUES  OF  MANUFACTURED  AND  INDUSTRIAL  PRODUCTS 


MANUFACTURED  ARTICLES 

1902 

1897 

DIFFERENCE 

Agricultural  implements  

82,075,609 

8    243  466 

Books,  maps,  etc  

988,195 

470,358 

Carriages  and  cars  

913,513 

80,065 

Copper  ingots  .    . 

198,438 

31  583 

Cotton  cloths    . 

385,086 

1  499  769 

Cotton  manufactures,  other  
Cycles  and  parts  of  . 

1,634,642 

98,476 

983,661 
339  563 

Builders'  hardware  

735,165 

377,549 

Sewing  machines  .             

182,710 

69,756 

Other  machinery  

894,330 

1,222,708 

Total  

OTHER  ARTICLES 

1902 

1897 

DIFFERENCE 

Corn  

$1  468  390 

$1  770  531 

Wheat 

3  769  577 

2  548  778 

Wheat  flour 

638  361 

2  415  519 

Coal   .      . 

5  473  177 

6  987  856 

Cotton              

4  509  205 

2  626  679 

Fruits  and  nuts 

1  345  °60 

566  584 

Furs  and  fur  skins  

667  164 

195,534 

Cot  ton  -seed  oil. 

261  688 

47  069 

Beef  salted  or  pickled 

240  978 

208  195 

Bacon  

557  827 

365  419 

Hams      .           

218  995 

188  116 

Total 

SUBTRACTION 


55 


3.  The  preceding  columns  show  the  values  of  the  different  kinds 
of  goods  exported  by  the  United  States  to  Canada  during  the  nine 
months  ending  March,  1902,  and  March,  1897,  respectively.  Fill 
out  the  vacant  column  of  differences,  marking  the  difference  + 
whenever  it  denotes  an  increase,  and  —  when  it  denotes  a  decrease. 

§36.  Data  for  Individual  Work. 

SCHOOL  STATISTICS  FOR  THE  THIRTY  LARGEST  CITIES  OP  UNITED  STATES 


CITY 

POPULA- 
TION. 

CENSUS 
1900 

POPULA- 
TION. 

CENSUS 
1890 

SCHOOL 
ENROLL- 
MENT. 
1900 

NUMBER 
TEACH- 
ERS. 
1900 

SCHOOL 
EXPENDI- 
TURES. 
1900 

New  York  

3,437,202 

2,492,591 

559,218 

12,212 

§21,040,810 

Chicago  

1,698,575 

1,099,850 

262,738 

5,951 

7,929,496 

Philadelphia 

1,293,697 

1,046,964 

151,455 

3,591 

4,677,860 

St.  Louis  
Boston  

575,236 
560,892 

451,770 

448,477 

82,712 
91,796 

1,751 

2,018 

1,526,140 
3,664,298 

Baltimore  

Cleveland  
Buffalo  

508,957 

381,768 
352,387 

434,439 

261,353 
255,664 

65,600 

59,635 
56,000 

1,600 

1,303 
1,300 

1,279,936 

,933,965 
,408,000 

San  Francisco 

342  782 

298,997 

48  517 

1  017 

152  631 

Cincinnati   
Pittsbur0"    . 

325,902 
321,616 

296,908 
238,617 

44,285 
50,000 

993 
1  000 

,064,047 

,757  381 

New  Orleans  
Detroit 

287,104 
285,704 

242,039 
205,876 

31,547 
40,303 

782 
966 

455,073 
1,251,825 

Milwaukee 

285,315 

204,468 

37,000 

900 

733,510 

Newark  

246,070 

181,830 

41,870 

851 

1,213,660 

Washington  
Jersey  City 

218,196 
206,433 

188,932 
163,003 

40,069 
32,174 

1,043 
586 

634,153 

Louisville  
Minneapolis  
Providence  

204,731 
202,718 
175,597 

161,129 
164,738 
132,146 

27,626 
38,591 
23,485 

650 
892 
682 

555,811 
841,000 
682,000 

Indianapolis  . 

169  164 

105  436 

27  334 

650 

729,106 

Kansas  City,  Mo.  .  . 
St.  Paul  
Rochester  

Denver  .    . 

163,752 
163,065 
162,608 

133  859 

132,716 
133,156 
133,896 

106  713 

28,280 
26,000 
24,896 

27  181 

700 
610 
692 

530 

524,065 
672,350 
682,018 

750  180 

Toledo  

131  822 

81  434 

21,467 

455 

471,314 

Allegheny  

129,896 

105  287 

20,104 

377 

835,634 

Columbus  
Worcester  
Syracuse  

125,560 
118,421 
108,374 

88,150 
84,655 
88,143 

18,855 
19,600 
21,090 

502 
574 

485 

771,132 
529,937 
409,073 

Total 

Problems  like  the  following  may  be  made  from  the  table,  and 
assigned  to  different  pupils. 


56  RATIONAL   GRAMMAR    SCHOOL    ARITHMETIC 

1.  Find  the  totals  of  all  columns  of  the  table  for  the  cities 
whose  inhabitants  numbered  over  500,000  at  the  1900  census. 

2.  Find  the  totals  for  the  cities  whose  populations  in  1900  were 
between  250,000  and  500,000;  for  cities  with  populations  between 
160,000  and  250,000. 

NOTE. — These  intervals  may  be  shortened  or  lengthened  at  will.  Indi- 
vidual pupils  may  do  different  parts  of  the  work,  thus  obtaining  a  large 
number  of  individual  problems.  In  each  case  the  pupil  should  tell  what 
his  total  means. 

3.  Find  the  increase  in  population  from  1890-1900  of  Cleve- 
land, Ohio ;  of  other  cities. 

4.  How  many  more  teachers  in  1900  were  there  in  Minneapolis 
than  in  Louisville?  in  Xew  York  City  than  in  Chicago?  in  Boston 
than  in  St.  Louis?  in  Chicago  than  in  Boston? 

5.  How  many  more  pupils  in  1900  were  enrolled  in  the  schools 
of  Boston  than  in  those  of  St.  Louis?   than  in  those  of  Balti- 
more? of  Cleveland? 

6.  How  much  more  money  was  expended  in  1900  on  schools 
in  Boston  than  in  St.  Louis?  than  in  Cleveland?     How  much  more 
in  Cleveland  than  in  Baltimore?   than  in  Buffalo?     How  much 
more  in  Pittsburg  than  in  Cincinnati? 

7.  How  do  the  combined  school  expenditures  for  New  York 
City,  Chicago,  Philadelphia,  and  Boston  compare  with  the  com- 
bined school  expenditures  of  all  the  rest  of  the  cities  in  the  table? 
How  do  those  of  New  York  City  and  Chicago  compare  with  the 
total  of  all  the  rest? 

8.  Find  the  totals  for  the  five  columns  of  the  entire  table. 
What  is  the  increase  in  the  total  population  of  these  cities  from 
1890  to  1900? 

§37.  Subtraction  of  Literal  Numbers. 

1.  How  many  five-cent  pieces  are  7  five-cent  pieces  — 3  five- 
cent  pieces? 

2.  How  many  dimes  are  12  dimes  —  5  dimes? 

3.  How  many  9's  are  16  9's -9  9's? 

4.  How  many  c's  are  13c-7c? 


MULTIPLICATION  5? 

5.  Write  the  differences  : 

183         26y        432         682         6830         1021s 


93  8y         37*         39z  97a  879* 


6.  A  lot  contains  1603  sq.  ft.,  and  the  house  covers  403  sq.  ft. ; 
how  many  square  feet  of  the  lot  are  not  covered  by  the  house? 

7.  A  boy  earned  153  cts.  on  Friday  and  spent  103  cts.  on  Sat- 
urday ;  how  many  cents  did  he  save? 

8.  Tell  what  number  x  stands  for  in  these  problems : 


(1)  12-a  =  7. 

(2)  65-a;  =  20. 
(3)  3-35=60. 
(4)  3-19-7. 

(5)  33-23  =  8. 

(6)  73-53  =  10. 
(7)  16^-133  =  15.  ' 
(8)  93-53  =  20. 
(9)  73=63. 
(10)  93  =  108. 

(11)  503  =  100. 
(12)  |3  =  50. 
(13)  |3=27. 
(14)  |3  =  7. 
(15)  f3  =  63. 

Literal  numbers  are  numbers  denoted  by  letters. 


MULTIPLICATION 
\.  Definitions.  ORAL  WORK 

1.  At  75^  a  day,  how  many  dollars  does  a  boy  earn  in  6  days? 

This  problem  may  be  solved  in  two  ways.     We  may  say  he 
earned  the  sum  of  75^  -f  75^  +  75^  +  75^  +  75$  -I-  75^,  which  is  $4.50. 
Or,  we  may  say  he  earned  6  times  75^  (6  x  75#),  or  $4.50. 
In  either  case  his  earnings  are  $4.50. 

2.  Which  is  the  shorter  way?     How  do  the  addends  compare 
in  the  first  solution? 

3.  Find  in  two  ways  how  far  a  vessel  will  sail  in  4  da.,  making 
22  mi.  a  day. 

4.  A  factory  employee,  working  by  the  hour,  makes  the  follow- 
ing daily  record  for  a  week :    8  hr. ;  *8  hr. ;   8-J-  hr. ;   9  hr. ;  9  hr. ; 
9£  hr.     How  many  hours  does  he  work  during  the  week? 

5.  Can  this  problem  be  solved  in  both  ways?     Give  reason  for 
your  answer. 

6.  When  the  addends  are  unequal,  as  in  problem  4,  what  is 
the  only  way  they  can  be  combined? 

7.  When  the  addends  are  equal,  as  in  the  first  two  problems,  in 
how  many  ways  can  they  be  combined?     Which  is  the  shorter 


58  RATIONAL   GRAMMAR   SCHOOL    ARITHMETIC 

way?     By  what  name  do  we  know  this  shorter  way?     What  then 
is  multiplication? 

Multiplication  of  whole  numbers  is  a  short  way  of  finding  the 
sum  of  equal  addends  when  the  number  of  addends  and  one  of 
them  are  given. 

The  given  addend  is  the  multiplicand. 

The  number  of  equal  addends  is  the  multiplier. 

The  result,  or  sum,  is  called  the  product. 

The  sign  of  multiplication  x  is  read  "multiplied  by"  when  it 
is  written  after  the  multiplicand  and  "times"  when  it  is  written 
before  the  multiplicand. 

Thus  22  mi.  x  6  =  132  mi.  is  read  "22  mi.  multiplied  by  G 
equals  132  mi.,"  and  0x22  mi.  =  132  mi.  is  read  "6  times  22 
mi.  equals  132  miles." 

An  expression  like  (5  x  22  mi.  =  132  mi.  is  called  an  equation. 

8.  At  $3.50  a  pair,  how  many  dollars  will  12  pairs  of  shoes 
cost? 

$3. 50  =  cost  of  1  pair; 
$3.50X  12  =  cost  of  12  pairs. 

$3.50  is  the  multiplicand,  12  is  the  multiplier ,  and  $42.00  is 
the  product. 

When  a  problem  is  expressed  in  an  equation,  the  equation  is 
called  the  statement  of  the  problem. 

§39.  WRITTEN   WORK 

Make  statements  and  solve : . 

1.  There  are   128  cu.  ft.  in  1  cd.  of  wood.     How  many  cubic 
feet  in  15  cords? 

STATEMENT.     15  x  128  cu.  ft.  =  ? 

Or,  15  X  128  cu.  ft.  =  a;  cu.  ft. 
Find  the  number  which  should  stand  in  place  of  x. 

The  number  which  should  stand  in  place  of  x  in  an  equation 
is  called  the  value  of  x. 

2.  There  are  5280  ft.  in  a  mile.     How  many  feet  in  24  miles? 

3.  12  Ib.  of  ham  at  170  per  Ib.  =  how  many  dollars? 

4.  A  cold  wave  from  the  northwest  traveling  at  the  rate  of  33 
mi.  an  hour  moves  how  many  miles  in  24  hours? 


MULTIPLICATION  59 

5.  If  galvanized  telegraph  wire  weighs  525  Ib.  per  mi.,  how 
many  pounds  of  wire  will  it  take  to  stretch  6  wires  from  Chicago 
to  Milwaukee,  a  distance  of  82  miles? 

6.  How  much  will  this  wire  cost  at  80  per  pound? 

7.  Find  the  cost  of  275  fence  posts  at  650. 

§.65  =  cost  of  one  post; 
275  =  number  of  posts ; 
$.  65  X  275  =  $178.75,  cost  of  27  posts. 

In     this    problem,     which 
number   is  the  multiplicand? 
325  Which  the  multiplier? 

455 
130 


$178.75 

But  it  is  usually  more  convenient  to  use  the  smaller 
number  as  the  multiplier.     This  may  be  done,  noticing 
that  if  each  post  cost  10,  275  posts  would  cost  $2.75;          1375 
but   as   each   post  costs   050,  the   cost   of  all  will  be         1650 
65  x  $2. 75,  and  the  work  may  be  arranged  as  shown 
here. 

Or,  we  may  say  275  times  $.65  is  the  same  as  65  times  $2.75, 
and  multiply  as  above. 

8.  In  64  pk.  there  are  how  many  quarts? 

9.  How  many  pounds  in  494  bu.  of  corn,  if  in  1  bu.  there  are 
56  Ib.?  how    many  pounds  in  25  bu.?  how  many  in  x  bushels? 

10.  There  are  231  cu.  in.  in  1  gal.     How  many  cubic  inches  in 
587  gall.?  in  y  gall.?  in  a  gallons? 

§40.  Tables. 

These  products  must  be  learned  thoroughly : 

3X3=9=3X3  3X4=  12  =  4X3 

4x3  =  12=    3X    4  4X4=16  =  4X    4 

5X3  =  15=3X5  5X4  =  20  =  4X5 

6X3  =  18=   3X    6  6X4=24  =  4X    6 

7x3  =  21  =3X7  7X4  =  28  =  4X7 

8X3  =  24  =3X8  8x4=32  =  4x8 

9x3  =  27  =3X9  9x4,=  36  =  4x9 

10  X  3  =  30  =  3  X  10  'iO  X  4  =  40  =  4  X  10 

11  X  3  =  33  =  3  X  11  11  X  4  =  44  =  4  X  11 

12  X  3  =  36  =  3  X  12  12  X  4  =  48  =  4  X  12 


60 


RATIONAL    GRAMMAR   SCHOOL   ARITHMETIC 


Learn  each  of  these  in  the  same  way: 


3X6  =  18 

3  X  7  =  21 

3  X  8  =  24 

3X9=  27 

4  X  6  =  24 

4  X  7  =  28 

4  X  8  =  32 

4X9=  36 

5  X  6  =  30 

5  X  7  =  35 

5  X  8  =  40 

5X9=  45 

6  X  6  =  36 

6  X  7  =  42 

6  X  8  =  48 

6X9=  54 

7  X  6  =  42 

7  X  7  =  49 

7  X  8  =  56 

7X9=  63 

8  X  6  =  48 

8  x  7  =  56 

8  X  8  =  64 

8x9=  72 

9  X  6  =  54 

9  X  7  =  63 

9  X  8  =  72 

9X9=  81 

10  X  6  =  60 

10  X  7  =  70 

10  X  8  =  80 

10  X  9  =  90 

11  X  6  =  66 

11  X  7  =  77 

11  X  8  =  88 

11  X  9  =  99 

12  X  6  =  72 

12  X  7  =  84 

12  X  8  =  96 

12  X  9  =  108 

ORAL   WORK 


1. 


At  current  prices,  find  the  cost  of  the  following  articles: 
|  Ib.  Oolong  tea.  2.    2  erasers. 


3  doz.  eggs. 
5  Ib.  ham. 
3  Ib.  leaf  lard. 

2  Ib.  Java  coffee. 

3  Ib.  best  quality  of  butter. 


3  tablets  linen  paper. 

3  packages  white  envelopes. 

4  doz.  pens. 

2  lead  pencils. 

2  bottles  writing  fluid. 


3.    2  doz.  oranges. 
2  doz.  lemons. 
1-J-  doz.  bananas. 
1  pk.  apples. 


4.   8  Ib.  rib  roast. 

2-J-  Ib.  porterhouse  steak. 
3  Ib.  lamb  chops. 
2  Ib.  sausage. 


5.    3  cd.  hard  wood. 
2  T.  hard  coal. 
1^  cd.  pine  slabs. 
4  T.  soft  coal. 


6. 


2  T.  hay. 

3  bu.  oats. 
2  bu.  corn. 

4  bales  straw. 


WRITTEN    WORK 


1.  Fill  out  the  vaoant  columns  and  find  the  totals  in  the  fol 
lowing  table  of  receipts  and  expenditures  for  the  fifty-acre  oat- 
field  which  was  shown  in  Fig.  7,  p.  11: 


MULTIPLICATION 


61 


In  Account  with  Fifty-Acre  Oatfield 
EXPENDITURES  RECEIPTS 


Mar.  20 

Removing  old  stalks  5  da. 

Feb.  15 

1800  bu.  oats  @  22J0 

@*1.25 

"      20 

85  loads  straw  @  $2.75 

"     20 

"     JiO 

100  bu.  seed  oats  @  350 
6i  da.   work  sowing  oats 

Expenditures    to    be    de- 

©$1.25 

ducted 

Aug.  18 

Harvesting  50  acres  oats 

Net  profit  from  50  acres 

@75<? 

"        "      per  acre 

"      18 

4J  da.  labor  in  harvesting 

©  $1.75 

Sept.  15 

Threshing  2148  bu.  @-  3<£ 

"      15 

5    da.    help    threshing  @ 

$1.50 

TOTAL 

Treat  the  meadow  account  similarly. 

In  Account  with  Ten-Acre  Meadow 
EXPENDITURES  RECEIPTS 


Nov.  20 
Aug.  30 
"     30 
"     30 

10    bu.    timothy    seed    @ 
$2.00 
Cutting    10   acres    hay  @ 
350 
Raking  and  shocking  hay 
5  da.  @  $1.50 
Stacking  15  tons  hay  ©550 

Dec.  15 

12  tons  hay  @  $0.50 

Expenses  to  be  deducted 
Net  profit  from  10  acres 
"        "      per  acre 

3.  Draw  up  the  following  items  into  an  account  like  the  one 
above;  and  find  totals  and  profit  or  loss  to  the  farmer. 
Day-Book  for  House  and  Barn  Lot  (10  acres}. 

1.  Apr.  15.  Bought  6  bu.  seed  po- 

tatoes @  §2 

2.  "     20.  Bought    30   pkg.  gar- 

den seeds  @  10j* 

3.  "     20.  Bought  4  pkg.  garden 

seeds  @  25?- 

4.  May  15.  Paid  5  da.   wages  @ 

§1.25 

25.  Sold  100  bunches  -on- 
ions @  2^ 

30.  Sold  148  bunches  cel- 
ery @  ty 

30.  Sold  200  bunches  let- 
tuce @  2^ 

30.  Sold  240  bunches  as- 


14.  July  31.  Paid  15  da.  wages  @ 

§1.50 

15.  "      31.  Sold  50  heads  cabbage 

@W 

16.  "      31.  Sold  20  doz.  cans  sweet 

corn  @  8,<^ 

17.  Aug.  31.  Sold  140  Ib.  tomatoes 


5. 

6. 

7. 
8. 

9.  June  30.  Sold  If  bu.  peas 

per  qt 
10. 

11. 

12.      "•    30.  Pa 


18. 
19. 
20. 


31.  Sold  10  bu.  early  apples 

@75^ 
31.  Sold  25  heads  cabbage 


per  qt. 
ld 


30.  Sold  28  bu.  green  beans 

@  40  per  qt. 
30.  Sold  300  bunches  as- 


paragus @ 
aid  18 
$1.25 


da.   wages  @ 


13.  July  31.  Sold  10  bu.  new  pota- 
toes @  II 


31.  Sold  15  bu.  peaches  @ 
W. 

21.  "      31.  10  doz.  ears  corn  @  7^ 

22.  "      31.  Paid  20  da.  wages  @ 

H.5Q 

23.  Sept.  30.  Sold  55  bu.  potatoes  @ 

65 

24.  "     30.  Sold  20  bu.  Lima  beans 

@  §1.25 

25.  "     30.  Paid  20  da.   wages  @ 

§1.25 

26.  Dec.   20.  Sold  30  bu.  apples  @§1 

27.  "      30.  Sold  10  bbl.   (40  gal. 

each)  cider  vinegar 
@  30^  per  gal. 


62  RATIONAL    GRAMMAR    SCHOOL    ARITHMETIC 

4.  Make  a  statement   from  these  items  of  general   expense, 
not  included  in  any  of  above  accounts,  and  find  the  total  profit  or 
loss: 

1.  Apr.  20.  Bought  3  sets  of  har-        7.  Aug.  30.  Built  corn   crib,    cost 

ness  @  $28  $112.00 

2.  "      25.  Paid  tax  on  160  acres  @        8.  Sept.  10.  Paid  for  120  mo.  pas- 

25/-  turing  stock  @  $1. 50 

3.  "      30.  Bought  1  wagon  @  860        9.  Feb.   28.  Paid  for  45  mo.    pas- 

4.  "      30.  Bought  2  plows  @  $35  turing  stock  @  $1.25 

5.  May   10.  Bought  3  cultivators  @      10.      "      28.  Paid  30  da.  fertilizing 

$25  @  $1.25 

6.  Aug.  20.  Paid  84  rods  tile  ditch-      11.      "      28.  Paid    8   da.    mending 

ing  @  50f;  fence  @  $1.25 

5.  Add  all  the  net  receipts  and  all  the  net  expenditures  for 
the  separate  fields  as  given  in  problems  1-4  here  and  in  §17  of 
the  Introduction.      Find  the  net  earnings  for   the    year  of   the 
whole  farm. 

§42.  Problems. 

1.  A  double  eagle  weighs  51 G  grains,  and  a  gold  dollar  25.8 
grains.     Find  the  difference  in  weight  between  a  double  eagle  and 
2  gold  dollars. 

2.  What  will  a  peck  of  peanuts  cost  at  5^'  a  pint? 

3.  At  $50  a  front  ft.,  what  will  50  ft.  of  city  land  cost? 

4.  If  milk  costs  6^  a  qt.  and  I  buy  2  qt.  a  day,  what  is  my  milk 
bill  for  April? 

5.  The  average  milk  yield  of  a  cow  was  6  qt.  a  da.  for  120  da. 
If  this  milk  was  all  sold  @  6^  a  qt.,  how  much  did  the  owner 
receive  for  it? 

6.  The  cow's  feed  cost  the  owner  $2.50  per  mo.  of  30  da. 
How  much  profit  did  the  owner  receive  from  the  cow's  milk  during 
the  120  days? 

7.  A  cow  gives  12  Ib.  of  milk  a  da.  and  the  butter  made  from 
this  milk  equals  ¥V  of  the  weight  of  milk.     How  many  pounds  of 
butter  does  the  milk  furnish  in  30  days? 

8.  If  butter  is  selling  for  35^  a  Ib.,  what  is  the  butter  yield  of 
this  cow  worth  in  30  days? 

9.  68  qt.  of  a  certain  Jersey  cow's  milk  yield  7  Ib.  of  butter. 
If  milk  is  selling  @  (}<f  and  butter  @  35^-,  is  it  more  profitable  to 


MULTIPLICATION  63 

sell  68  qt.  of  the  milk  of  this  cow  as  milk  or  as  butter?     How 
much  more  profitable  is  it? 

10.  The  pulse  of  a  healthy  man  beats  72  times  per  minute. 
The  heart  contracts  once  for  each  pulse  beat.     How  many  times 
do  the   muscles  of   a  healthy  man's  heart  contract  in  1  hr.?  in 
1  da.  at  the  same  rate? 

11.  Count  yonr  own  pulse  beats  for  1  min.  and  find  how  many 
times  your  heart  beats  in  a  day  at  this  rate. 

12.  Tie  a  heavy  ball  to  a  fine  string,  or  split  a  bullet  and 
fasten  it  to  a  thread,  and  suspend  the  string  to  a  hook  or  nail 
so  that  1^  ft.  of  it  may  vibrate  freely.     Start  it  swinging  and 
count  the  number  of  vibrations  per  minute.     At  the  same  rate 
how  many  times  would  the  ball  vibrate  in  24  hours? 

13.  Solve  the  same  problem  with  the  suspending  string  3  ft. 
long;  with  the  string  3  ft.  3  in.  long. 

14.  My  watch  ticks  270  times  per  minute;   how  many  times 
does  it  tick  in  1  hr.?  in  1  day? 

15.  Light  travels  186,600  mi.  per  second;  how  far  does  it  travel 
in  I  day? 

16.  A  man  smokes  3  cigars  a  day,  and  pays  15^  apiece  for 
them.     How  much  does  this  add  to  his  expense  account  in  365 
days? 

§43.  Multiplying  by  Factors. 

1.  7x8-56  28x2  =  56  14x4  =  56 
7,  8,  28,  2,  14,  and  4  are  all  factors  of  56. 

2.  Write  the  factors  of  44,  48,  96,  35,  84,  81,  49,  108. 

3.  The  two  equal  factors  of  144  are  12  and  12.     Give  the  two 
equal  factors  of  4,  9,  25,  49,  36,  121. 

4.  8x2  =  ?         8x2x5  =  ?         8x10  =  ? 

5.  12x2  =  ?        12x2x3  =  ?        12  x  6  =  ? 

6.  20x4  =  ?        20x4x3  =  ?        20x12  =  ? 

7.  25x4  =  ?        25x4x6  =  ?        20x24  =  ? 

8.  64x9  =  ?        64x9x7  =  ?        64x63  =  ? 

9.  Instead  of  multiplying  a  number  by  15,  by  what  two  num- 
bers in  succession  may  I  multiply  it  and  get  the  same  product? 


64  RATIONAL    GRAMMAK    SCHOOL    ARITHMETIC 

10.  Multiplying  any  number  by  21  gives  the  same  product  as 
multiplying  it  by  what  two  numbers  in  succession? 

11.  56  x  2  x  4  gives  the  same  product  as  56  multiplied  by  what 
single  number? 

12.  Multiplying  a  number  by  several  factors  one  after  another 
gives  the  same  product  as  multiplying  it  by  what  single  number? 

A  number  which  has  factors  other  than  itself  and  1  is  called 
a  composite  number.  A  number  such  as  3,  5,  17,  etc.,  which  has 
no  factors  other  than  1  and  itself,  is  a,  prime  number. 

13.  Instead  of  multiplying  a  number  by  a  composite  number, 
in  what  other  way  may  I  multiply  it  to  obtain  the  same  product? 

14.  Which  of  these  numbers  are  prime,  and  which  composite: 
24,  25,  19,  6,  2,  21,  23,  84,  45,  43,  27,  23? 

§44.  Multiplying  by  10,  100,  1000,  10,000. 

1.  84x10  =  ?      84x100  =  ?       84x1000  =  ?       84x10,000  =  ? 

2.  327x10  =  ?    327x100  =  ?     327x1000  =  ?     327x10,000  =  ? 

3.  How  may  any  whole  number  be  quickly  multiplied  by  10, 
100,  1000,  or  10,000? 

§45.  Multiplying  by  a  Number  Near  10,  100,  1000,  10,000. 

1.  746x9=?      . 

SOLUTION.— This  is  once  746  less  than  10  X  746,  or  7460  -  746  =  6714. 

2.  965  x      99  =  ?     Ans.  96,500-    965  =  95,535. 

3.  965  x   98  =  ?  Ans.  96,500 - 1930  =  94,570. 

4.  965  x  101  =  ?  Ans.  96,500+  965  =  97,465. 

5.  965  x  102  =  ? 

6.  873  x   11  =  ? 

7.  8473  x  1001  =  ? 

8.  8473  x  999  =  ? 

§46.  Multiplying  by  25,  50,  12^,  75,  500,  or  250. 

1.  65  x  25  =  ?  25  =  i  of  100.  ±  of  65  x  100  =  1625. 

2.  83  x  50  =  ?  50  =  i  of  100.  |  of  83  x  100  =  4150. 

3.  638  x  12i  =  ?  12|  =  i  of  100.  -J  of  638  x  100  =  7975. 

4.  132  x  75  =?  75-=fof  100.  f  of  132  x  100  =  9900. 

5.  640  x  500  =  ?  500  =  |  of  1000. 

6.  740  x  250  =  ?  250  =  •  of  1000. 


MULTIPLICATION  65 

We  see  from  problem  1  that  to  multiply  by  25,  we  may  multi- 
ply by  100  and  take  £  of  the  product. 

7.  Make  a  rule,  different  from  the  ordinary  rule,  for  multiplying 
quickly  by  50;  by  12|;  by  250;   by  500. 

8.  Make  a   rule  for    multiplying    quickly   by   33£;  by   333£. 
This  shows  the  importance  of  remembering  that : 

|  of  100  =  50  ^  of  1000  =  500 

i  of  100  =  25  \  of  1000  =  250 

|  of  100  =  75  a  of  1000  =  750 

4  of  100  =  12£  |  of  1000  =  125 

|  of  100  =  37*  |  of  1000  =  375 

|  of  100  =  62^  |  of  1000  =  625 

I  of  100  =  87*  I  of  1000  =  875 

/e  of  100  =    6i  &  of  100°  =  62a 

i  of  100  =  33*  i  of  1000  =  3331 

|  of  100  =  661  §  of  1000  =  6661 

9.  Make  rules  for  multiplying  quickly  by  the  numbers  on  the 
right  hand  side  of  these  equations. 

§47.  Multiplying  When  Some  Digits  of  the  Multiplier  Are  Factors 
of  Others. 

1.  From  twice  a  number,  how  may  you  obtain  4  times  the  same? 
From  4  times  a  number,  how  obtain  8  times  the  same? 

2.  Multiply  19,279  by  842. 

SOLUTION.—       19279 

842 

38558  =  19279  X  2,  the  first  partial  product 
771160  =  20  times  the  first  partial  product 
15423200  =  20  times  the  second  partial  product 

16232918 

3.  Observe  the  relations  of  the  digits  of  each  multiplier  below, 
and  shorten  the  work  of  multiplication,  as  above.    Prove  work : 

4,783x363  29,847x248 

12,875x442  86,729x393 

§48,  Checking  Multiplication. 

Any  of  the  above  ways  may  be  used  to  test  the  correctness  of 
products  obtained  in  the  usual  way. 


6(5  RATIONAL    GRAMMAR    SCHOOL    ARITHMETIC 

To  check  by  casting  out  the  nines,  cast  the  nines  out  of  the 
multiplicand,  the  multiplier,  and  the  product.  Multiply  the 
excesses  of  the  multiplicand  and  the  multiplier.  Cast  the  nines  out 
of  this  product.  If  this  last  excess  equals  the  excess  of  the 
original  product,  the  work  is  probably  correct. 

ILLUSTRATION.—  To  check  6848X619  =  4238912,  excess  in  6848  =  8; 
excess  in  619  =  7;  excess  in  4238912  =  2;  7x8  =  56,  excess  in  56  =  2. 
As  excess  in  4238912  and  in  56  are  equal,  the  work  is  checked. 

A  very  useful  check  against  blunders  is  to  examine  the  facts  of 
the  problem  and  to  decide  about  what  the  answer  must  be,  before 
beginning  to  solve  it. 

§49.  Multiplying  by  Fractional  Numbers. 

1.  How  many  square  yards  in  a  room 
4  yd.  x  8-J-  yards? 

SOLUTION.—  How  many  sq.  yd.  fill  a  strip 
1   yd.    wide  extending  along  the  long  side? 


How  many  such  strips  cover  the  floor?    4  X 
.    W 


FIGURE  25  sq.  yd.  =#  sq.  yd.    What  is  £C? 

2.  What  is  the  cost  of  68£  acres  of  land  @  $80? 

SOLUTION.—  68  X  §80  =  $5440,  and  |  of  $80  =  §540.    Then  68i  x  §80  = 
$5480.     Observe  that  68  \  X  80  means  68  X  80,-f  \  of  80. 

3.  Tell  what  these  problems^mean  and  then  solve  them: 
$64  x  16|  =  ?  81  Ib.  x  25J  =  ?      24  ft.  x  8f  =  ? 
5280  mi.  x  14|  =  ?     897  yd.  x  19J  =  ?      5  J-  sq.  yd.  x  6  =  ? 
160  A.  x  12f  =  ?        5£  sq.  yd.  x  5£  =  ? 

NOTE.—  Solve  the  last  problem  by  drawing  a  square  f>£  units  long  and 
5J  units  wide,  and  dividing  it  up  into  square  units. 

The  problems  just  solved  show  that  f  x  8  means  f  of  8,  or  3 
of  the  4  equal  parts  of  8. 

4.  Give  the  meaning  of  these  problems  and  find  the  value  of 
the  letter  in  each  : 

f  x  9  =  x.     f  x  21  =  x.     £  x  56  =  y.     ft  x  108  =  y.     if  x  75  =  z. 

§50.  Suggestions  for  Problems. 

Make  and  solve  problems  based  on  the  following  facts  : 
1.  A  steel  rail  weighs  64  Ib.  per  yd.  of  length.     The  distance 
from  Washington  to  Baltimore  is  42  mi.  ;   to  Philadelphia,  138 
mi.  ;  to  New  York,  227  mi.  ;  to  Boston,  459  miles. 
NOTE  —There  are  1760  yd.  in  1  mile. 


MULTIPLICATION  67 

2.  There  are  double  tracks  between  Washington  and  each  of 
the  cities  mentioned. 

3.  The  distance  from  New  York  to  San  Francisco  is  3262 
miles. 

4.  The  distances  in  miles  from  Chicago  to  14  railroad  centers 
of  the  United  States  are  given  here : 

Indianapolis 184  Rochester 605 

St.  Louis 283  Baltimore 801 

Cincinnati 298  Washington 820 

Cairo 364  New  Orleans 912 

St.  Paul 410  New  York 913 

Omaha 490  Denver 1028 

Buffalo 536  San  Francisco 2349 

5.  Galvanized  telegraph  wire  weighing  572  Ib.  per  mi.  is  used 
for  distances  over  400  mi.,  and  for  distances  under  400  mi.,  wire 
weighing  378  Ib.  per  mi.  is  used. 

6.  8  wires  run  between  Chicago  and  New  York. 

7.  Galvanized  iron  telegraph  wire  costs  6^-  per  pound. 

8.  Sound  travels  in  water  at  the  rate  of  4708  ft.  per  second; 
in  air  1130  ft.  per  second. 

9.  Silver  is  worth  50^  an  oz.,  12  oz.  to  the  pound. 

10.  Hay  costs  $23  per  T. ;   oats  42^  per  bu.      A  horse  con- 
sumes 162  Ib.  of  hay  and  10  bu.  of  oats  a  month.     Bedding  costs 
$2  a  month. 

11.  Standard  silver  is  f9¥  pure  silver  and  TV  copper.     A  silver 
dollar  weighs  412.5  grains.     The  total  number  of  silver  dollars 
coined  on  this  basis  was  378,166,769. 

12.  The  number  of  grains  in  an  ounce  of  silver  is  480.     The 
amount  of  silver  bought  under  the  Sherman  Law  by  the  United 
States  government  for  coinage  purposes  was  168,674,682  ounces. 

13.  Light  travels  186,600  mi.  per  second.     It  requires  448  sec- 
onds for  light  to  reach  the  earth  from  the  sun. 

14.  It  reaches  the  earth  from  the  moon  in  1^\  seconds. 

15.  An  oz.  of  pure  gold  is  worth  $20.67.     There  are  12  oz.  in 
1  Ib.  of  gold. 

16.  An  Alaskan  miner  can  take  away  200  Ib,  from  the  mining 
district. 


68 


RATIONAL    GRAMMAR    SCHOOL    ARITHMETIC 


17.  A  cu.  ft.  of  granite  weighs  170  Ib.     A  granite  step  meas- 
ures y  x  2'  x  8'. 

18.  The  distance  around  the^  driving  wheel  of  a  locomotive 
engine  is  22  ft.     In  going  a  certain  distance  the  driver  turned 
5280  times. 

19.  In  the  year  1901,   82,305,924  Ih.  of  tea  and  [511,041,459 
Ih.  of  coffee  were  imported  into  the  United    States.     Tea  was 
worth  48^  and  coffee  26^  per  pound. 

20.  A  prize-winning  steer  weighed  15.03  cwt.  (1  cwt.=  100  Ib.) 
and  sold  for  $9.00  per  hundredweight. 

21.  Another  steer  of  the  same  lot  weighed  1622  Ib.  and  sold 
for  $8.85  per  hundredweight. 

22.  A  boy^bicyclist  rode  a  miles  per  da.  for  ~b  days. 

23.  Pupils  should  prepare  and  solve  problems  based  upon  price 
lists  obtained  from  the  grocer  and  the  butcher  (see  p.  8),  or  from 
the  market  reports  of  the  daily  papers. 

§51.  Rainfall. — How  long  is  a  cubic  inch  (Fig.  26)?  how  wide? 
how  high?     What  is  a  cubic  foot? 
A  cubic  yard? 


FIGURE  26 


FIGURE  27 


A  tin  box  3  in.  square  on  the  bottom  and  9  in.  high  (Fig.  27) 
was  used  by  a  school  as  a  rain  gauge.     A  second  tin  box  1  in. 
the  bottom  arid  9  in.  high  was  used  to  measure  the 


square  on 

depth  of  water  in  the  large  box. 


After  the  water  was  poured 


MULTIPLICATION 


69 


1  foot 


from  the  large  box  into  the  small  box,  the  depth  of  water  in  the 
small  box  was  measured  with  a  thin  stick  and  a  foot  rule. 

1.  The  rain  gauge  was  placed  one  evening  where  the  rain  could 
fall  freely  into  its  open  top.     During  the  night  it  rained  and  the 
next  morning  the  water  was  poured  from  the  gauge  into  the  small 
box.     It  filled  the  small  box  to  a  depth  of  9  in.     How  deep  did 
the  water  fill  the  large  box?     How  many  cubic  inches  of  water 
were  caught  in  the  large  box? 

One  cu.  in.  of  water  for  every  sq.   in.   of  surface  is  what  is 
meant  by  1  in.  of  rainfall.     What  is  6  in.  of  rainfall? 

2.  How  many  cu.  in.  of  water  are  there  in  a  layer  1  in.  deep 
in  the  large  box  (Fig.  27)?    2  in.  deep?    5  in.  deep?    9  in.  deep? 

3.  How  many  cu.  in.  of  water  fell  on  1  sq. . 
ft.   of  the  ground  during  a  rainfall  of  1  in. 
(Fig.  28)?  of  2  in.?  of  3  in.?  of  6  inches? 

The  number  of  cubic  units  (cu.  in.,  cu.  ft., 
cu.  yd.,  etc.),  a  vessel  holds,  when  full,  is  called 
its  capacity. 

4.  What    is    the    capacity    of    a    square- 
cornered  box  3"  x  3"  x  9"? 

5.  What  was  the  depth  of  rainfall  during  a 

shower  if  the  large  box  caught  enough  water  to  fill  the  small  box 
half  full?  to  a  depth  of  3  in.?  of  6  in.?  of  1  in.?  of  2  in.?  of  7 
inches? 

6.  How  many  cu.  in.  of  water  fell  on  1  sq.  ft.  of  the  ground 
during  a  shower  giving  1  in.  of  rainfall?  2  in.?  -J  inch? 

7.  During   a  rainfall  of    1  in.   how  many  sq.  ft.   of   ground 
received  enough  water  to  make  1  cu.  ft.?    2  cu.   ft.?    12  cubic 
feet? 

8.  During  June,  1902,  the  rainfall  in  the  vicinity  of  Chicago 
was  6£  in.     During  this  month  how  many  cu.  in.  of  water  fell 
on  1  sq.  in.  of  ground?  on  1   sq.  ft.?   on  12  sq.  ft.?   on  1  sq. 
yd.?  on  30  sq.  yd.?  on  \  sq.  yd.?  on  30^  square  yards? 

9.  Just  before  a  shower  set  an  uncovered  bucket  where  the 
rain  may  fall   freely  into   it.      After   the   shower  measure  the 
depth  of  the  water  in  the  bucket  to  find  the  depth  of  rainfall. 


FIGURE  28 


70  RATIONAL    GRAMMAR    SCHOOL    ARITHMETIC 

Find  how  many  cu.  in.  or  cu.  ft.  of  water  fell  on  each  sq.  ft.? 
each  sq.  yd.?  each  sq.  rd.  of  the  ground? 

NOTE. — The  bucket  used  must  have  the  same  size  at  the  top  and  bot- 
tom, with  straight  sides.     Why? 

10.  During  the  first  9  mo.  of  the  year  1902,  the  region  about 
Chicago  received  32  in.  of  rainfall.     How  many  cu.  ft.  of  water 
fell  during  this  time  on  a  garden  bed  10'  x  14'?  on  a  garden  48'  x 
84'?  over  a  city  block  250'  x  350'? 

11.  If   the  walls  of  your   schoolroom  were  water-tight,    how 
many  cu.  ft.  of  water  would  it  hold  if  it  were  filled  1  ft.  deep? 
2  ft.  deep?  5  ft.  deep?  8  ft.  deep?  to  the  ceiling? 

|  12.  Find  the  areas  of  these  rectangles : 

6"x8",  12"x28",  40'x64', 
375' x  486',  9'xz',  9yd.  xzyd., 
a  mi.  x  1)  mi.,  x  units  x  y  units. 

How  can  you  find  the  number  of  square  units  in  any  rectangle? 
NOTE.— x  X  y  is  written  scy  and  read  "a?,  y." 

13.  How  many  cubic  feet  are  there  in  1  cu.  yd.?  in  18  cu.  yd.? 

14.  What  is  the  capacity  of  a   square-cornered  box  3"  x  3" 
x  9"?  3'  x  3'  x  9'?  of  a  square-cornered  room  or  space,  3  yd.x  3yd. 
x  9  yd.?  3  rd.  x  3  rd.  x  9  rd.?  3x3x9?  3  x  3  x  a? 

15.  Find  the  capacity  of  a  square-cornered  box  3"  x  4"  x  7"; 
3"x4"x8";    3'x4'x8';    3'x4'xl5';    3  yd.  x  3  yd.  x  3  yd. ;  3 
units  x  4  units  x  15  units;  3  x  4  x  #;  3  xxxy,  a  units  x  b  units 
x  c  units. 

NOTE.— The  product  x  X  y  X  z  is  written  xyz  and  read  "x,  y,  z." 

16.  How  can  you  find  the  number  of  cubic  units  in  any  square- 
cornered  vessel? 

17.  Find  the  total  weight  of  a  snow  load  of  25  Ib.  per  sq.  ft. 
on  a  flat  rectangular  roof  25'  x  48'.     Find  the  weight  of  a  load 
of  15  Ib.  per  square  foot. 

18.  Find  the  total  weight  of  the  shingles  and  sheathings  for 
both  sides  of  the  roof  shown  in  Fig.  29,  if  shingles  weigh  3  Ib. 
per  sq.  ft.,  and  sheathing  5  Ib.  per  square  foot. 

NOTE. — The  eaves  project  2  ft.  over  the  plate. 


MULTIPLICATION 


71 


19.  Find    the  weight    of   the  snow    load  on   both   sides   of 
the  roof,  the  weight  on  each  sq.  ft.  of  surface  being  12  pounds. 

20.  Find  the  cost  at  3^  per  sq.  yd.  of  lathing  and  plastering 
the    four  walls   and    the   ceiling, 

no  allowances  being  made. 

21.  What   is   the   area  of   the 
square  ABCD  of  Fig.  29?     What 
is  the  area  of  the  triangle  DEC? 

22.  A   strong    gale,    giving    a 
pressure   of    16    lb.    per    sq.    ft., 
blows  squarely  against  the  end  of 

the  building.     What  is  the  total  FIGUBE  29 

wind  pressure  against  the  end  including  the  gable? 

23.  How  many  cu.  ft.  of  space  are  inclosed  by  the  walla  to 
the  base,  DC,  of  the  gables? 


D    24'     C 
A             B 

§52.  Algebraic  Problems. 

4  x  x  is  written  4#.     If  4z  =  24,  what  is  the  value  of  x? 

Notice  that  the  equation  4#  =  24  is  the  statement  of  this  prob- 
lem. A  boy  reads  24  pages  of  a  book  in  4  days;  how  many  pages 
does  he  read  a  day?  What  does  x  itand  for  in  this  problem? 

1.  What  number  does  x  stand  for  in  these  equations: 

(l)3z=18.  (4)z-  8  =  15.  (7)z  +  9  =  17.  (10)  5x  +  %x  =  70. 
(2)  Qx  =  48.  (S)a?-18=  4.  (8)*+  6  =  22.  (11)  9z-  3x  =  36. 
(3)82=72.  (6)z-  9=  8.  (9)^  +  16  =  25.  (12)  Bx  +  5x  =  26. 


2.  Write  the  sum  of  a  and  b. 

3.  Write  the  difference  of  a  and  b. 

4.  Write  the  product  of  a  and  />;  of  a  and  b  and  c;  of  x  and 
y  and  z. 

5.  Answer  the  following  questions  if  a  =  9,  b  =  8  and  c  =  6  : 
4rt  =  ?     12J  =  ?     9c  =  ?     cib  =  ?     4ab  =  ?      abc  =  ?     Sabc  =  ? 

6.  In  the  shortest  way  you  can,  write  eight  times  or;  seven  times 
y  ;  twenty-five  times  a  times  b  ;  a  times  x  times  ?/. 


72  RATIONAL    GRAMMAR    SCHOOL   ARITHMETIC 

7.  In  the  shortest  way,  write  and  read  a  times  b\  c  times  x\ 
fifteen  times  a  times  b  times  x. 


DIVISION 
§53.  Division  and  Subtraction  Compared.— ORAL  WORK 

1.  A  man  owes  a  debt  of  $12  which  he  is  to  discharge  by  work 
at  $2  per  day.     How  much  will  he  owe  at  the  end  of  the  first  day? 
of  the  second  day?  of  the  third  day?  of  the  fourth  day?  the  fifth? 
the  sixth? 

2.  How  long  will  it  take  to  cancel  the  debt? 

3.  How  many  times  may  2  be  subtracted  from  12,  leaving  no 
remainder?     How  many  2's  are  there  in  12? 

4.  A  man  buys  a  horse  for  $90  and  is  to  pay  $15  a  month 
until  the  horse  is  paid  for.     How  much  does  the  man  owe  after 
the  first  payment?    after  the  second?    the  third?   fourth?   fifth? 
sixth? 

5.  How  many  months  will  it  take  to  pay  for  the  horse? 

6.  How  many  times  may  15  be  subtracted  from  90,  leaving  no 
remainder?     How  many  15's  in  90? 

7.  What  is  one  of  the  6  equal  parts  of  12?  of  90? 

8.  A  rectangular  plot  of  ground  8  yd.  wide  by  18  yd.  long  is 
covered  with  bluegrass  sod.     How  many  square  yards  of  sod  does 
it  contain?     After  a  strip  of  sod  1  yd.  wide,  extending  the  length 
of   the  plot,  has  been  removed,  how  many  square  yards  of  sod 
remain? 

9.  How  many  square  yards  remain  after  the  removal  of  2  such 
strips?  of  3?  of  4?  of  5?  of  6?  of  7?  of  8? 

10.  How  many  times  may  18  sq.  yd.  be  subtracted  from  144 
sq.  yd.,  leaving  no  remainder?     How  many  18's  are  there  in  144? 
What  is  one  of  the  8  equal  parts  of  144? 

WRITTEN    WORK 

1.  There  are  160  bricks  in  a  pile;  find  by  subtraction  how 
many  loads  of  20  bricks  each  there  are  in  the  pile.  How  many 
20's  are  there  in  160? 


DIVISION  73 

2.  Find  by  successive  subtraction  how  many  88 's  there  are  in 
610.     What  is  one  of  the  7  equal  parts  of  016? 

3.  Find  by  subtraction  how  many  months  of  30  da.  there  are  in 
270  da.     What  is  one  of  the  0  equal  parts  of  270? 

4.  What  number  may  be  subtracted  4  times  in  succession  from 
120,  leaving  no  remainder? 

5.  Find  by  subtraction  how  many  times  $1203  is  contained  in 
$5052. 

6.  Tell  how  to  find  by  subtraction  how  many  times  one  num- 
ber is  contained  in  another. 

7.  Find  in  a  shorter  way  how  many  12's  there  are  in  156. 

S.  Find  in  a  shorter  way  than  by  subtraction  one  of  the  14 
equal  parts  of  224.  By  what  name  do  you  know  this  short  way? 

Division  is  a  short  way  of  finding: 

(1)  One  of  a  given  number  of  equal  parts  of  a  number. 

(2)  How  many  equal  parts  of  a  given  size  there  are  in  a  given 
number. 

When  the  term  "division"  has  the  meaning  of  (2),  the  process 
to  which  it  applies  may  be  called  measurement. 

In  the  case  of  whole  numbers,  division  is  a  short  way  of  sub- 
tracting one  number  from  another  a  certain  number  of  times  in 
succession. 

§54.  Division  and  Multiplication  Compared. — ORAL  WORK 

1.  A  horse  traveled  72  mi.  in  8  hr. ;    find  the  number  of  miles 
traveled  per  hour. 

2.  4  bu.  potatoes  cost  $2.40.     What  was  the  price  per  bushel? 

3.  A  room  4  yd.  wide  contains  24  sq.  yd. ;  what  is  the  length 
of  one  side? 

4.  15  Ib.  sugar  cost  90^;  what  is  the  price  per  pound? 

5.  At  $0  per  ton,  how  many  tons  of  coal  can  be  bought  for 
$180? 

6.  A  floor  containing  132  sq.  ft.  is  11  ft.  wide;    what  is  the 
length? 

7.  A  train  runs  420  mi.  in  12  hr. ;  find  the  average  number  of 
miles  per  hour. 

8.  Find  the  cost  of  8  Ib.  veal  at  IZ<p  per  pound. 

9.  8  Ib.  veal  cost  96^;  what  was  the  price  per  pound? 


74  RATIONAL    GRAMMAR   SCHOOL    ARITHMETIC 

10.  I  paid  96^  for  veal  at  12^  per  pound;   how  many  pounds 
did  I  buy? 

11.  What  is  the  cost  of  3  T.  hay  at  $12  per  ton? 

12.  At  $12  per  ton,  how  many  tons  of  hay  can  be  bought 
for  $36? 

13.  A  man  paid  $36  for  3  T.  hay;  what  was  the  price  per  ton? 

14.  Find  the  cost  of  25  bbl.  (barrels)  flour  at  $5. 

15.  Paid  $125  for  25  bbl.  flour;  what  was  the  price  per  barrel? 

16.  Paid  $125  for  flour- at  $5;  how  many  barrels  were  bought? 

17.  The  two  factors  of  a  number  are  13  and  7;   what  is  the 
number? 

18.  The  product  of  two  numbers  is  91  and  one  of  the  numbers 
is  13 ;  what  is  the  other  number? 

19.  The  product  of  two  numbers  is  240  and  one  of  the  num- 
bers is  20 ;  what  is  the  other? 

20.  20  x  x  =  240 ;  what  number  does  x  stand  for? 

21.  Tell  what  number  the  letter  stands  for  in  each  of  these 
equations : 

12#  =  96;  Sx  =  56;  la  =  56;  95  =  72;  10a  =  150;  24m  =  240. 
Division  is  a  way  of  finding  one  of  two  numbers  when  their 
product  and  the  other  number  are  given. 
The  product  is  called  the  dividend. 
The  given  number  is  the  divisor. 
The  required  number  is  the  quotient. 

ILLUSTRATION.  — The  product  of  two  factors  is  490,  and  one  of  the  fac- 
tors is  7.  What  is  the  other  factor? 

70 
7)490 

We  may  also  say  that  the  dividend  is  the  number  to  be 
divided,  the  divisor  is  the  number  by  which  the  dividend  is 
measured  or  divided,  and  the  quotient  is  the  measure. 

The  sign  +  of  division  is  read  "divided  by,"  as  60  +  12  =  5. 

60  -f-  12=  5  may  also  be  written  in  the  following  ways : 

_5  12)60(5  M  =  5 

12)60  j>0  12 

The  first  and  second  are  in  common  use  in  division. 
In  the  third,  the  line  placed  between  two  numbers  shows  that 
the  number  above  it  is  to  be  divided  by  the  number  below  it ;  as 


DIVISION  75 

in  J,  1  is  the  dividend  and  3  the  divisor.    The  fraction  itself  is  the 
quotient.     The  line  between  the  two  numbers  is  the  division  sign. 
The  fourth  sign  is  called  the  solidus  (sol'i-dus). 

§55.  Short  Division. 

1.  How  many  gallons  are  there  in  296  pints? 

SOLUTION. — As  there  are  8  pt.  in  1  gal.  there  are  as  many  37 

gallons  in  296  pt.  as  there  are  8's  in  296.  g  )~296~ 

2.  How  many  days  in  120  hours? 

3.  At  $8  per  ton,  how  many  tons  of  coal  can  be  bought  for 
$240? 

4.  984  marbles  are  distributed  equally  among  a  certain  number 
of  boys.     Each  boy  has  82  marbles.     There  are  how  many  boys? 

5.  Selling  at  G  for  a  cent,  how  much  will  a  dealer  receive  for 
540  marbles? 

6.  A  man  paid  $2.60  for  4  bbl.  of  lime.     What  did  each  bbl. 
cost? 

7.  I  bought  6  Ib.  of  butter  for  $1.62.     What  was  the  price  per 
pound? 

8.  A  dressmaker  used  84  yd.  of  cloth. for  6  dresses,  allowing 
the  same  amount  for  each  dress.     How  many  yards  in  each? 

9.  A  train  ran  150  mi.  in  6  hr.     Not  allowing  for  stops,  what 
was  the  average  number  of  miles  per  hour? 

Find  the  value  of  x  in  problems  10  and  11 : 

10.  48  qt.  =  x  gallons. 

11.  At  6^  I  can  buy  x  Ib.  of  sugar  for  $1.50. 

12.  9  papers  of  needles  cost  72^.     One  paper  costs  how  many 
cents? 

13.  In  1901,  the  number  of  trains  entering  Chicago  every  24 
hr.  was  about  1320;  what  was  the  average  number  per  hour? 

ORAL    WORK 
What  does  x  stand  for  in  each  of  the  following  equations? 

1.  63  +  7  =  z  4.     45  +  3  =  z  7.   120  +  3  =  x 

2.  630  -*•  9  =  x  5.     96  +  8  =  x  8.   108  -*-  9  =  x 

3.  48  •*-  4  =  x  6.   960  +  8  =  x  9.     72  -*•  6  =  x 


76  RATIONAL   GRAMMAR   SCHOOL   ARITHMETIC 

When  dividend  and  divisor  are  small  numbers,  the  quotient  is 
readily  seen.  We  say  it  is  obtained  by  inspection. 

When  the  divisor  is  a  large  number,  the  quotient  is  not  readily 
found  by  inspectipn. 


§56.     Applications.  WRITTEN   WORK 

1.  In  51,448  qt.  how  many  pecks  are  there? 

SOLUTION. — In  51,448  qt.  there  are  as  many  pecks  as  there  are  8  qt.  in 
51,448  quarts. 

8  is  contained  in  51,000,  6000  times,  with  a  remainder  of  3000    Write 
6  in  the  thousands  place  in  the  quotient.     3  thousands  =  30  hundreds;  30 
hundreds  and  4  hundreds  =  3400.    8  is  contained  in  3400,  400 
6431     times,  with  a  remainder  of  200.     Write  4  in  the  hundreds 
8 )  51448     P^ce  in  the  quotient.     200  =  20  tens ;  20  tens  and  4  tens  =  24 
tens.    8  is  contained  in  24  tens  3  tens  times,  without  a  remain- 
der.    Write  3  in  the  tens  place  in  the  quotient.     8  is  contained  in  8  units 
1  unit  time.     Write  1  in  the  units  place  in  the  quotient. 
In  51,448  qt.  there  are  6431  pecks. 
Check:  6431  X  8  =  51,448. 

2.  There  are  5280  ft.  in  a  mile ;  how  many  yards  are  there  in 
1  mi.?  in  2  miles? 

3.  If  limestone  weighs  160  Ib.  per  cubic  foot,  how  many  cubic 
feet  are  there  in  a  piece  of  limestone  weighing  0400  pounds? 

4.  If  marble  weighs  170  Ib.  per  cubic  foot,  find  the  number  of 
cubic  feet  in  a  piece  of  marble  weighing  5100  pounds. 

5.  If  sand  weighs  120  Ib.  per  cubic  foot,  find  the  number  of 
cubic  feet  in  a  load  of  sand  weighing  4800  pounds. 

6.  A  train  of  12  sleeping  cars  is  840  ft.  long.     If  the  cars  are 
all  the  same  length,  how  long  is  each  car? 

7.  A  steel  rail  30  ft.  long  weighs  720  Ib. ;   what  is  its  weight 
per  yard  of  length? 

8.  An  iron  beam  24  ft.  long  weighs  1080  Ib. ;   what  is  the 
weight  of  a  piece  of  the  beam  1  ft.  long? 

9.  A  steel  girder  weighs  1728  Ib.     Each  foot  of  length  weighs 
48  Ib.     How  long  is  it? 

10.  There  were  487,918   foreign  immigrants    to  the  United 
States  in  the  year  1901.      What  was   the  average  number  per 
month?  per  day?  (30  da.  =  1  month.) 


DIVISION 


77 


11.  During  1900  there  were  448,572  immigrants.     Find  the- 
average  number  per  month. 

12.  Answer  the  same    question  for  1891,   1892,   1893,   1895, 
and  1897,  the  numbers  for  these  years  being  560,319;  623,084; 
502,917;   258,530,  and  25JO,S32. 

13.  In  Albany,  N.   Y.,  there   are   30  mi.   of  street  railway, 
operated  by  (100  men.     What  is  the  average  number  of  employees 
per  mile? 

14.  From  the  data  here  given  answer  the  same  question  for 
these  cities: 


CITY 

MILES 

EMPLOYEES 

St   Joseph   Mo            ...    . 

35 

175 

Memphis    Tenn              

70 

490 

Oakland    Cal                      

80 

560 

Hartford    Conn.  .             

33 

660 

Worcester,  Mass  

43 

473 

Peoria   111 

50 

275 

15.  Find  the  value  of  x  in  each  case: 

(1)  3264-    G  =  ff        (4)   89,705+    5  =  x       (7)         24,568+    8  =  x 

(2)  9432  +  12  =  x        (5)  78,870  +  11  -  a       (8)         45,900  +  12  =  a; 
(3)2247-    7  =  x        (0)75,699-    9  =  a       (9)   1,241,196  -  11  =  x 

16.  How  can  you  prove  the  correctness  of  your  work  in  divi- 
sion? 

17.  Find  what  x  equals  in  these  equations: 


!2-io 


W 


x 
66°° 


4A°  =  49 

X 

!E.9o 

a; 


,       640 
(8)^  =  66 


§57.  Long  Division. 

When  dividend  and  divisor  are  both  large  numbers,  it  becomes 
necessary  to  show  all  the  steps  of  the  work. 


78 


RATIONAL   GRAMMTAR   SCHOOL   ARITHMETIC 


1.   14,487  -H  33  =  ? 


30  V  =  439,  quotient 
400 


33)14487 

13200     =  400  X  33 


1287 
990     =    30 


297 

297     =     9  X  83 


SHORTER  FORM 

439,  quotient 

divisor,  33)14487,  dividend 
132 


128 
99 


297 


SOLUTION.  — Beginning  at  the  left  of  the 
dividend;  33  is  not  contained  in  1,  nor  in 
14,  but  it  is  contained  in  14,400,  400  times. 
Write  the  400  above  the  dividend  and 
subtract  400  X  33  =  13,  200  from  the  divi- 
dend, leaving  1287.  This  remainder  must 
also  be  divided  by  33. 

33  is  not  contained  a  whole  number  of 
times  in  1,  nor  in  12,  but  it  is  contained 
30  times  in  1280.  Write  the  30  above  the 
400  over  the  dividend  and  subtract  30  X  33 
=  990  from  1287,  leaving  297. 

33  is  contained  in  297,  9  times,  leaving 
no  remainder. 

Thus  we  see  33  is  contained  in  14,487 
400  -f  30  -f-  9  =  439  times. 

The  work  may  be  shortened  a  little  by 
omitting  the  zeros  and  writing  numbers  in 
the  shorter  form  below. 


Check:   439  X  33  =  14,487. 
division  is  probably  correct. 


Since  this    is  the  given    dividend,    the 


When  a  zero  appears  in  the  quotient  proceed  as  follows : 
2.  18,722-46  =  ? 


407 


46)18722 
184 

322 

322 


SOLUTION. — Begin  as  above.  46  is  contained  in  187,  4  times, 
with  the  remainder  3.  Bring  down  the  2  in  the  dividend, 
giving  32.  46  is  not  contained  in  32  a  whole  number  of  times. 
Write  0  in  tens  place  in  the  quotient  and  bring  down  the 
next  2  of  the  dividend,  giving  322.  46  is  contained  in  322, 
7  times.  Write  the  7  in  units  place  in  the  quotient. 


Check:  407  X  46  =  18,722,  which  equals  the  dividend. 


Complete  these  equations  and  check  your  work : 

3.  3,580-45=  7.  33,768-72  = 

4.  15,552  -  64  =  8.   35,096  -  82  = 

5.  18,144-56=  9.  62,328-84  = 

6.  20,088-72=  10.  44,928-96  = 


DIVISION  79 


>58.  Exercises. 


1.  There  are  52  wk.   in  a  year.     My  friend  is  1872  wk.  old; 
how  many  years  old  is  he? 

2.  There  are  160  sq.  rd.  in  1  A.  and  a  farmer  pays  75^  per  acre 
for  cutting  and  binding  wheat.     How  much  will  it  cost  to  cut  and 
bind  the  wheat  on  a  field  68  rd.  by  80  rods? 

3.  A  farmer  paid  a  man  $21.00  to  shock  his  wheat,  wages 
being  $1.50  per  day.     How  many  days  did  the  man  work? 

4.  In  1880^25  farm  wagons  sold  for  $2250  and  in  1900  15 
such  wagons  sold  for  $855.     How  much  less  was  the  average  cost 
of  a  farm  wagon  in  1900  than  in  1880? 

5.  In  1880  a  Minnesota  farmer  paid  $3900  for  12  twine  bind- 
ers and  in  1900  14  twine  binders  cost  him  $1680.     How  much  had 
the  average  price  of  twine  binders  fallen  during  these  20  years? 

6.  A  steel  rail  weighing  72  Ib.  per  yard  is  30  ft.  long.     How 
many  men  are  needed  to  carry  it,  each  man  carrying  90  pounds? 

7.  In  25  da.  a  man  earned  $56.25  husking  corn  at  3^  per 
bushel.     How  many  bushels  per  day  did  he  husk? 

8.  A  city  lot  175  ft.   long,  containing  8750  sq.  ft.,  sold  for 
$6000.     If  the  short  side  fronts  the  street  what  was  the  price  per 
foot  of  frontage? 

9.  The  force  required  to  draw  a  street  car  on  a  level  track  is 
35  Ib.  per  ton  (2000  Ib.)  of  the  combined  weight  of  the  car  and 
its  load.     What  force  is  needed  to  draw  a  car  weighing  5600  Ib. 
when  it  is  loaded  with  60  passengers  whose  average  weight  is  140 
pounds? 

10.  At  the  speed  of  an  ordinary  horse  car  a  horse  can  exert 
about  125  Ib.  of  force  in  drawing  the  car.     How  many  horses  will 
be  needed  to  draw  the  car  of  problem  9,  no  horse  to  draw  more 
than  125  pounds? 

11.  At  a  slow  walk  a  horse  can  exert  about  330  Ib.  of  force. 
The  force  required  to  draw  a  loaded  wagon  on  a  level  pavement 
is  -fy   of  the   weight   of   the  wagon  and  load.     A   coal  wagon 
weighing  4860  Ib.  is  loaded  with  4  T.  of  coal.     How  many  horses 
will  be  needed  to  draw  the  load  over  a  level  pavement,  no  horse 
drawing  more  than  330  pounds? 


80 


RATIONAL   GRAMMAR    SCHOOL    ARITHMETIC 


12.  A  horse  can  exert  1540  Ib.  of  force  for  a  few  minutes. 
A.  box  car  weighing  30  T.  is  loaded  with  32  T.  The  force  needed 
to  move  the  loaded  car  is  -fa  of  the  combined  weight  of  the  car 
and  load.  How  many  horses  will  be  needed  to  start  the  car  on 
a  level  track? 

§59.  Larger  Numbers. 

1.  5,128,672-9272  =  ? 


SOLUTION.— The  divisor,  9272,  being  too  large  to  use 
readily,  we  first  use  a  trial  divisor.  For  the  same 
reason,  we  select  a  trial  dividend.  92,  the  trial  divisor, 
is  contained  in  512,  5  times;  but  as  the  whole  divisor  con- 
tains four  digits  the  partial  dividend  must  be  enlarged. 
9272  is  contained  in  51,286,  5  times.  The  quotient  figure  5 
is  of  the  same  order  as  the  last  figure  of  the  trial  dividend, 
which  is  hundreds.  We  write  5  in  hundreds  place  in  the 
quotient.  Multiplying  the  whole  divisor  by  the  quotient 
figure  we  have  the  product  46,360.  Subtracting  this  prod- 
uct from  the  trial  dividend,  4926  remains.  4926  hundreds 
=  49,260  tens ;  and  49.260  tens  +  7  tens  =  49,267  tens.  Con- 
tinue in  the  same  manner  with  each  step  that  follows. 
The  last  subtraction  gives  a  remainder  of  1256.  This  remainder  must 


553 

9272)5128672 
46360 

49267 
46360 

29072 
27816 

1256 


also  be  divided  by  the  divisor  9272. 

5,128,672  -h  9272  = 


9272 
553 

27816 
46360 
46360 

5127416 
1256 

5128672 


The  553  is  the  whole,  or  integral  par,;  of  the  quotient  and 
the  £|ff  is  the  fractional  part. 


Check:  Multiply  the  divisor  by  the  quotient,  and  to  the 
product  add  the  remainder.  The  result  should  equal  the 
dividend. 


Solve  the  following  problems  and  check  your  work  : 

2.131,320-536  =          4.       630,861-2731  = 
3.  195,936-624=          5.   1,057,536-4352 


§60.  Geography. 

1.  From  the  table  of  §30  the  area  of  Massachusetts  is  seen  to 
be  8315  sq.  mi.,  and  that  of  Illinois  is  56,650  sq.  mi.  How  many 
states  the  size  of  Massachusetts  could  be  made  from  Illinois? 


DIVISION  81 

2.  From  the  same  table  the  area  of  New  England  is  found  to 
be  66,465  sq.  mi.     How  many  states  as  large  as  New  England 
could  be  made  from  Texas? 

3.  Make  and  solve  other  problems  like  these,  using  the  table. 

4.  The  same  table  shows  the  area  of  Connecticut  to  be  4990 
sq.  mi.  and  its  population  for  1900  to  be  908,420.     How  many 
persons  per  square  mile  are  there  in  Connecticut? 

NOTE. — In  problems  such  as  this,  where  the  fractional  part  of  the 
quotient  has  no  meaning,  drop  the  remainder  if  it  is  less  than  half  the 
divisor,  and  add  one  unit  to  the  whole  part  of  the  quotient  if  the  remain- 
der is  more  than  half  of  the  divisor. 

5.  The  table  of  §35  gives  the  population  of  Connecticut  for 
1890  as  740,258.     What  was  the  population  of  Connecticut  per 
square  mile  in  1890? 

6.  Answer  questions  4  and  5  for  your  own  state. 

7.  From  the  same  table  answer  questions  4  and  5  for  Okla- 
homa territory. 

8.  Make   and  solve  similar  problems  for  any  states  you  are 
studying  in  your  geography. 

9.  The  area  of  Switzerland  is  15,781  sq.  mi.  and  its  popula- 
tion is  2,933,334.     What  is  the  population  of  Switzerland   per 
square  mile? 

10.  640  A.  =  1  sq.  mi.      How  many  square  miles  in    534,528 
acres? 

11.  The  area  of  the  state  of  Texas  is  265,780  sq.  mi. ;  that  of 
New  Jersey  is  7815  sq.  mi.     How  many  states  the  size  of  New 
Jersey  could  be  made  from  Texas?     How  many  the  size  of  Delaware, 
which  contains  2050  square  miles? 

12.  Porto  Eico  has  an  area  of  3531  sq.  mi.,  and  a  population 
of  953,243.     How  many  inhabitants  does  it  support  to  the  square 
mile?  • 

13.  The  greatest  ocean    depth    found  is  31,614  ft.   near  the 
island  of  Guam,  in  the  Pacific  ocean.     The  highest  mountain  in 
the  world  is  Mt.  Everest  in  Asia,  which  rises  29,002  ft.  above  sea 
level.     Find  the  difference  of  level  in  miles  between  the  greatest 
ocean  depth  and  the  greatest  land  altitude. 


RATIONAL    GRAMMAR    SCHOOL    ARITHMETIC 


14.  The  state  of  New  York,  with  an  area  of  49,170  sq.  mi., 
supports  a  population  of  7,268,894.     How  many  inhabitants  does 
it  average  to  the  square  mile? 

15.  Hawaii  has  an  area  of  6449  sq.  mi.,  and  a  population  of 
154,001.    Find  the  average  per  square  mile. 

16.  In  1891  the  total  wheat  area  of  North  and  South  Dakota 
was  4,882,157  A.;  the  yield  was  81,819,000  bu.     What  was  the 
average  yield  per  acre? 

17.  In  the  year  1900  Kentucky  had  22,488   A.  of  rye  under 
cultivation.     The  total  yield  was  292,344  bushels.      What  was  the 
average  yield  per  acre? 

18.  In  the  year  1890  there  were  210,366  persons  employed  in 
manufacturing  in  Chicago.     The  total  wages  paid  amounted  to 
$123,955,001.     What  was  the  average  wage  paid  to  each  person? 

19.  In   1880,    3519   factories    in    Chicago    together    yielded 
$249,022,948  worth  of  products.     In  1890,  9977  factories  yielded 
$577,234,446  worth.     During  which  year  was  the  average  product 
per  factory  greater,  and  by  how  much? 

20.  A  comparison  of  the  density  of  population  (population  per 
square  mile)  may  be  obtained  by  finding  the  density  of  population 
of  these  countries: 


COUNTRY 

AREA 

POPULATION 

DENSITY 

German  Empire.  .       ... 

208,830 
120,979 
115,903 
204,092 
110,646 
8,660,395 
3,688,110 

56,345,014 
41,454,578 
26,107,304 
38,641,333 
32,449,754 
135,000,000 
76,212.168 

Great  Britain  

Austria 

France 

Italy 

Russia 

United  States  

NOTE. — Only  the  whole  numbers  need  be  found  for  these  quotients. 

§61.  Division  by  Multiples  of  10.— ORAL  FORK 

1.  Multiply  each  of  these  numbers  by  10: 

568          1268          306          $86.50         $8.65 

2.  Multiply  each  of  the  same  numbers  by  100 ;  by  1000. 

3.  Divide  each  of  these  numbers  mentally  by  10 : 

5680          12680         3060         $865.00         $86.50 


DIVISION  83 

4.  Make  a  rule  for  dividing  any  number  quickly  by  10 ;  by  100 ; 
by  1000 ;  by  1  with  any  number  of  zeros  after  it. 

5.  Name  these  quotients  orally: 

60  + 10  =  ?  6  +  2  =  y  60  +  20  =  y 

860  +  10  =  ?  86  +  2  =  ?  860+20  =  ? 

$64.20  +  10  =  ?         $6.42  +  2  =  ?         $64.20  +  20  =  y 

Examining  your  answers  to  the  questions  just  asked,  make  a 
rule  for  dividing  a  number  quickly  by  20;  by  200;  by  2000. 

6.  180  +  10  =  ?  18  +  3  =  ?  180  -  30  =  y 
630  +  10  =  y             63  +  3  =  y             630  +  30  =  ? 

From  these  answers  make  a  rule   for   dividing   any  number 
quickly  by  30;  by  300;  by  3000. 

7.  Make  a  rule  for  dividing  a  number  quickly  by  40;  by  400; 
by  4000;  by  50;  by  800;  by  1200;  by  1500. 

8.  Make  a  rule  for  dividing  any  number  quickly  by  any  whole 
number  of  tens,  as  40,  70,  90,  160;  by  any  whole  number  of  hun- 
dreds; of  thousands. 

9.  Cutting  off  zero  from  the  right  of  a  number  has  what  effect 
on  the  numbery  cutting  off  2  zerosy  3  zerosy 

66 -f-  10  =  61e0,  or  6.6. 
165  H-  10  =  16&,  or  16.5. 

75 +-100=,%.  or  .75. 
478 +•  100  =  4-^,  or  4. 78. 

10.  Using  first  10  and  then  100  as  a  divisor,  give  and  show  the 
quotients  of  the  following : 

400  500  2200  3300 
460  790  4280  4860 
287  439  9647  5732 

11.  There  are  10  pk.  in  a  barrel.     How  many  barrels  in  1488 
pecksy 

12.  There  are  60  Ib.  in  a  bushel  of  potatoes.   How  many  bushels 
in  486  poundsy 

13.  100  lb.=  1  cwt.     How  many  hundredweight  in  825  poundsy 

14.  200  Ib.  pork  =  1  bbl.     How  many  barrels  in  7624  pounds? 


84  RATIONAL   GRAMMAR   SCHOOL   ARITHMETIC 

15.  How  many  minutes  in  42GO  seconds? 

SOLUTION.—  When  there  are  ciphers  at  the  right  of  both  71 

dividend  and  divisor,  cut  off  an  equal  number  of  ciphers 
from  both  and  divide.  60)48601 

1C.  How  many  barrels  will  be  needed  for  4800  Ib.  of  beef, 
allowing  200  Ib.  to  the  barrel? 

§62.  Other  Methods  of  Shortening  Division. 

1.  Solve  these'problems: 

5000)25000  500)2500          50)250          5)25 

How  do  the  quotients  compare?  the  dividends?  the  divisors? 

2.  Remembering  that  the  dividends  stand  above  the  line  and 
the  divisors  below,  solve  these  problems  : 

36  12 


81  27  ~  9^  3 

How  do  the  quotients  compare?  the  dividends?  the  divisors? 

3.  How  do  the  quotients,  the  dividends,  and  the  divisors  com- 
pare in  these  problems  : 

256  64  =  9  16  4 

128  32"  8  2 

4.  To  divide  324  by  81  what  smaller  numbers  may  I  use  to  get 
the  same  quotient?     How  can  I  obtain  these  smaller  numbers  from 
324  and  81? 

SOLUTION.—  We  see  that,  as  324  =  27X12  and  81  =  27x3,  we  may 
write 

13  X  27  _  12  _  . 

3  X  27  ~~  ~3~  = 
This  can  be  indicated  thus  : 

12_Xjfr_ 

sxzr 

Eemoving  these  factors  is  called  cancellation.     It  can  often 
be  used  to  simplify  the  division  of  products. 

5.  Answer  the  same  questions  for  25G  +  128. 

Any  factor  of  both  dividend  and  divisor  may  be  dropped  or 
stricken  from  both  and  the  remaining  factors  divided. 


DIVISION  85 

6.  Solve  these  problems  by  cancellation : 

34x16  _  9  6x8x3  =  9  15  x  21  x  846  =  9 

2x16  "  ^  '  2  x  8  x  3  ~  3  x    7  x  846  ~ 

18x3x4x67  =  ?     (5)625  =  ?     (6)  ^  =  ?     (7)-  =  ? 
&  x  4  x  o  x  07  <*o  loJ  o4 

To  use  cancellation  effectively,  methods  of  finding  factors  of 
numbers  are  necessary. 

§63.  Tests  of  Divisibility. 

1.  Which  of  these  numbers  are  exactly  divisible  (can  be  exactly 
divided)  by  2 : 

12       24       36       23      45      18      37      40       59       61 
What  are  the  last  digits  of  the  numbers  which  2  will  divide? 

TEST  FOR  THE  FACTOR  2 :  If  a  number  ends  in  0,  2,  4,  6,  or  8,  it  can  be 
exactly  divided  by  2. 

2.  Which  of  these  numbers  are  exactly  divisible  by  10: 

*   24     30     45     50     700     640     83     765     6400 
What    is    the    last   digit    of    the    numbers    which    10    exactly 
divides? 

TEST  FOR  THE  FACTOR  10:  If  a  number  ends  in  0,  10  exactly  divides  it. 

3.  Make  a  rule  for  testing  whether  100  divides  a  number. 

4.  Which  of  these  numbers  does  5  exactly  divide: 

16     18     35     25     60     20     28     65     460     675     1260 
What  are  the  last  digits  of  the  numbers  which  5  will  divide? 

TEST  FOR  THE  FACTOR  5:   If  a  number  ends    in  0  or  5,  5  exactly 
divides  it. 

5.  Which  of  these  numbers  does  3  exactly  divide: 

12     17     24     81     27     93     64     75     126     324     185 
Of  the  numbers  which  3  exactly  divides,  will  3  also  exactly  divide 
the  sum  of  the  digits?     Of  the  numbers  3  does  not  exactly  divide, 
is  the  sum  of  the  digits  exactly  divisible  by  3? 

TEST  FOR  THE  FACTOR  3 :    If  3  exactly  divides  the  sum  of  the  digits 
of  a  number  it  divides  the  number  also. 

6.  Of  these  numbers  what  ones  does  9  exactly  divide: 

126     368     453     729     819     639     2358 


86  RATIONAL    GRAMMAR    SCHOOL    ARITHMETIC 

See  whether  9  will  exactly  divide  the  sum  of  the  digits  of 
the  numbers  that  9  exactly  divides. 

TEST  FOR  THE  FACTOR  9 :  If  9  exactly  divides  the  sum  of  the  digits 
of  a  number  it  also  exactly  divides  the  number. 

7.  Which  of  these  numbers  does  4  exactly  divide: 

113     124     368     560     375     486     1204 

See  whether  of  the  numbers  it  exactly  divides  4  also  exactly 
divides  the  number  indicated  by  the  last  two  digits.  For 
example,  in  the  third,  368  can  be  exactly  divided  by  4  and  so  also 
can  68. 

TEST  FOR  THE  FACTOR  4 :  If  the  number  denoted  by  the  last  two 
digits  of  a  number  can  be  divided  by  4,  the  entire  number  can  be  exactly 
divided  by  4. 

8.  Which  of  these  numbers  are  divisible  by  25 : 

60  175  285  625  1350  1275  8645  8675  8625  8650 

25  divides  175  exactly  and  25  also  exactly  divides  75,  which  is  the 
number  denoted  by  its  last  two  digits.  Is  this  true  of  all  numbers 
25  exactly  divides?  Is  it  true  of  any  numbers  25  does  not  exactly 
divide? 

TEST  FOR  THE  FACTOR  25:  If  the  number  denoted  by  the  last  2  digits 
of  any  number  is  divisible  by  25  the  entire  number  is  divisible  by  25. 

TEST  -FOR  THE  FACTOR  6 :  test  for  both  2  and  3. 

TEST  FOR  ANY  COMPOSITE  FACTOR:  test  singly  for  all  the  factors  of 
the  composite  factor. 

9.  TEST  FOR  DIVISIBILITY  by  such  numbers  as  36,  216,  27, 
49,  etc.,  which  contain  some  factor  two  or  more  times. 

10.  Pick   out   the   numbers   of    this   list    which   are   exactly 
divisible  by  2 : 

6     81     65     72     86     129     9864     8643     7986     16,835      29,860 

11.  Pick  out  those  which  can  be  exactly  divided  by  3;   by  9; 
by  4;  by  5;  by  10;  by  12;   by  15. 

§64.  Checking  Division. 

Division  may  be  checked  by  multiplying  the  divisor  by  the 
quotient  and  adding  the  remainder  to  the  product.  If  the  sum 
equals  the  dividend,  the  work  is  checked. 


DIVISION  8? 

Another  check  is  to  divide  by  the  factors  of  the  divisor  suc- 
cessively and  note  whether  the  final  quotient  is  the  same  as  that 
given  by  the  complete  divisor. 

To  check  by  casting  out  the  nines,  add  the  excess  in  the 
product  of  the  excesses  of  divisor  and  quotient  to  the  excess  of  the 
remainder.  If  the  excess  of  this  sum  equals  the  excess  of  the 
dividend,  the  division  is  probably  correct. 

ILLUSTRATION. — Check  the  work  of  problem  1,  §59. 

Cast  the  9's  out  of  the  dividend,  5,128,672.     The  excess  is  4- 

Cast  the  9's  out  of  the  divisor,  9272.     The  excess  is  2. 

Cast  the  9's  out  of  the  quotient,  553.     The  excess  is  4. 

Cast  the  9's  out  of  the  remainder,  1256.     The  excess  is  5. 

The  product  of  the  excesses  of  divisor  and  quotient  is  8. 

The  excess  of  8  is  8  itself.  Add  this  8  to  the  excess  of  the  remainder, 
giving  13. 

The  excess  of  this  13  is  4.  and  as  this  equals  the  excess  4  of  the  divi- 
dend, the  division  is  probably  correct. 

It  is  even  more  important  in  division  than  in  multiplication 
to  examine  a  problem  carefully  before  beginning  to  solve  it.  Try 
to  foresee  about  what  the  answer  must  be.  This  often  avoids 
blunders. 

ILLUSTRATION. — 1.  If  it  requires  480  slates  to  cover  a  square  (1 00  sq.  ft.) 
of  roof  surface,  how  many  squares  are  there  in  a  roof  which  requires 
13,680  slates  to  cover  it? 

Pupil  should  at  once  notice  that  if  it  required  500  slates  to  cover  a 
square,  there  would  be  a  little  more  than  13,680  -f-  500,  or  136  -*-  5  =  27 
squares  and  he  might  guess  28  squares.  The  actual  division  of  13,680  by 
480  gives  28.5  squares. 

First  form  a  rough  estimate  of  the  answer  and  then  solve  these 
exercises :  • 

2.  $238  was    paid  for  flour   @  $3.50;   how  many  bbl.  were 
bought? 

SUGGESTION. — How  many  bbl.  would  there  have  been  if  the  price  had 
been  $7.00  a  barrel? 

3.  In  still  air  a  hawk  flew  375  miles  in  2J  hr.     What  was  its 
speed  per  hour? 

4.  In  3.9  hr.  a  crow  flew  97.5  m. ;  find  the  rate  of  flight  per 
hour. 

5.  From  a  certain  cow  -fa  °f  ^ne  milk  was  butter-fat.     The 
cow  gave  12  Ib.  of  milk  per  day.     How  many  pounds  of  butter-fat 
will  the  milk  from  this  cow  yield  in  70  days? 


88 


RATIONAL    GRAMMAR    SCHOOL    ARITHMETIC 


§65.  Applications  of  Cancellation. 

In  all  the  problems  of  this  list  indicate  your  divisions  and 
multiplications,  then  apply  the  tests  of  divisibility,  and  cancel 
the  factors  found  in  both  dividend  and  divisor.  Multiply  the 
uncanceled  factors  in  the  dividend  together  and  divide  this  product 
by  the  product  of  the  uncanceled  factors  of  the  divisor. 

ILLUSTRATION.  —  A  speed  of  102  mi.  per  hour  equals  how  many  feet 
per  second?  Indicate  the  work  thus: 

34      22 

102  X  5280  _  V&  X  #g_  34  X  22  _  748 
60  X  60  M  5         =   5   = 


Ans.  149g  ft.  per  second. 

1.  It  took  225,280  Ib.  of  steel  rails  to  lay  1  mi.  of  single-track 
railroad.     What  was  the  average  weight  per  yard  of  the  rails? 

2.  Find  the  number  of  cubic 
feet  in  a  ton  of  2000  Ib.  for  each 
of   the  substances   given   in  the 
table. 

3.  The     average     speed     of 
American  express  trains  is  about 
35    mi.    per    hour   for  long  dis- 
tances.    How  many  feet  per  sec- 
ond is  this? 

NOTE.—  60  sec.  =  1  min.  ;  60  min.  =  1  hour. 
SOLUTION.  —  Put  work  in  this  form: 

7       22 
35  X  5280      35  X  &f      154      r 


LB.  PEK 
C'u.  FT. 

Cu.  FT. 
PEK  T. 

Granite  .... 
Limestone  .  . 
Marble  
Sandstone.  . 
Slate  

170 
160 
170 
140 
170 

Cast  Iron  .  .  . 
Steel 

450 

480 

60x60 


r1  ,   ., 

Ans-  51i  ft  per  second- 


3 


4.  In  both  England  and  America,  the  average  speed  of  express 
trains  for  distances  from  100  to  250  mi.  is  about  40  mi.  per  hour. 
How  many  feet  per  second  is  this? 

5.  For  long  distances  the  average  speed  of  English  express 
trains  is  about  43  mi.  per  hour.     How  many  feet  per  second  is 
this? 

6.  In  1893  an  express  train  in  the  United  States  ran  1  mi.  at 
the  rate  of  98  mi.  per  hour.     How  many  feet  per  second  is  this? 


DIVISION  89 

7.  A  railroad  train  ran  for  1  min.  at  the  speed  of  130  mi.  per 
hour.     Find  the  number  of  feet  per  second  which  it  traveled. 

8.  Sound  travels  in  air  at  the  rate  of  about  750  mi.  per  hour. 
How  many  feet  per  second  is  this? 

9.  Cannon  balls  have  been  thrown  at  a  speed  of  810  mi.  per 
hour  for  a  few  seconds.     This  is  how  many  feet  per  second? 

10.  The  moon  moves  around  the  earth  at  a  speed  of  about 
1,840,000  mi.  in  30  da.     How  many  feet  is  this  per  second? 

NOTE.— 24  hr.  =  1  day. 

11.  In  365  da.  the  earth  moves  around  the  sun  through  a 
distance  of  about  584,000,000  mi.     What  is  the  earth's  speed  in 
miles  per  second? 

12.  The   earth,  by  turning   on   its   axis,  carries   a  place  on 
its  equator  about  25,000  mi.  in  24  hours.     How  many  feet  is  it 
carried  per  second? 

§66.  The  Lever. 

1.  Rest  a  foot  rule  on  a  support  as  at  F  in  Fig.  30  and  load  it 
with  G  oz.  at  W.     What  weight  at  P  will  balance  the  foot  rule? 

2.  If  the  rule  is  first  balanced*  on  the  three-inch  mark  as  in 
Fig.  31  how  many  ounces  will  be  needed  at  P  to  balance  18  oz. 
at  W? 

P  F          W 

I    I    I    I    i    i    I    I 


P 

F 

W 

III!) 

11111 

1    1    1 

FIGURE  30  FIGURE  31 

3.  If  the  support  were  placed  at  the  two-inch  mark  and  the 
rule  balanced  how  many  ounces  at  P  would  balance  24  ounces  at 
IT? 

The  point  that  the  (foot  rule)  bar  rests  on  is  called  the 
fulcrum. 

Call  the  distance  from  the  fulcrum,  F,  to  the  weight,  W,  to  be 
balanced  FW,  Fig.  32,  and  the  distance  from  the  fulcrum  to  the 
power,  P,  needed  to  balance  the  weight  FP. 

*  A  piece  of  brick  or  other  substance  should  be  placed  on  the  short  end  of  the  rule  or 
bar  to  balance  it  exactly  with  the  long  end,  in  all  these  problems. 


90  RATIONAL    GRAMMAR    SCHOOL    ARITHMETIC 

4.  If  FP  is  just  as  long  as  FW>  20  Ibs.  at  P  will  balance  how 
many  pounds  at  W? 

W 
P  F  W 


I 


FIGURE  32  FIGURE  33 

5.  If  FP  is  twice  as  long  as  FW,  10  Ib.  at  P  will  balance 
how  many  pounds  at  Wt 

6.  A  light  bar  of  convenient  length,  like  the  one  shown  in 
Fig.   32,  was  supported  as  at  the  point  F.     The  bar  was  then 
balanced  by  placing  a  small  piece  of   brick  on  the  short  end. 
A  weight  was  then  placed  on  the  end  W  of  the  balanced  bar  and 
the  weight  needed  to  balance  W  was  hung  at  P. 

The  distances  FW  and  FP  were  then  measured  on  a  number 
of  different  bars  similarly  balanced  and  loaded  and  the  following 
table  was  made: 

AtW  FW  AtP  FP 

8  oz.  3  in.  3  oz.  8  in. 

10  oz.  2  in.  2  oz.  10  in. 

12  oz,  4  in.  6  oz.  8  in. 

22  oz.  1  in.  2  oz.  11  in. 

8  Ib.  6  in.  G  Ib.  8  in. 

12  Ib.  3  in.  4  Ib.  9  in. 

7  Ib.  4  in.  2  Ib.  14  in. 

Compare  the  products  of  the  numbers  in  columns  1  and  2  with 
the  products  of  those  of  columns  3  and  4.  What  do  you  find? 

7.  When  a  lever  balances,  the  load  multiplied  by  its  distance 
from  the  fulcrum  equals  what  other  product? 

8.  In  Fig.  33  if  IF  =  120  Ib.,  FW=  I  ft.,  and  FP  =  4  ft.,  what 
force  at  P  will  just  balance  the  stone  at  W?     What  force  would 
be  needed  if  FW  =  G  in.,  and  FP  =  6  feet? 

9.  If  the  load  is  W  Ib.,  the  power  P  Ib.,  the  distance  from 
F  to  W  is  w  feet  and  from  F  to  P  is  p  feet,  what  equation  can 
you  write  if  the  bar  balances? 

NOTE.— W  X  w  is  written  Ww. 


DIVISION 


91 


10.  By  what  must  5  be  multiplied  to  equal  6  x  10? 

SOLUTION. — Call  x  the  number  by  which  5  must  be  multiplied,  then: 

Statement,  5x  =  6  X  10,  and 


=  12. 


Check:  5X12  =  6x10  =  60. 

11.  Letting  p  be  the  distance  from  F  io  P,  find  what  p  is  in 
these  problems,  representing  conditions  for  a  balanced  lever,  can- 
celing when  you  can  and  checking  all  answers  : 

FW  =  2  ft.          TF=    80  Ib.         P  =  16  Ib.          FP  =p  ft. 
FW=6in.         TF=500lb.         P  =  25  Ib.  FP  =  p  in. 

FW=3it.         JF  =  420lb          P=211b.          FP  =  p  ft. 

12.  The  edge  of  a  sheet 
of  tin  was  placed  in  the  shears 
as  in  Fig.  34.  If  fa  =  2  in., 
fb  =  8  in.,  and  a  pressure  of 
22  Ib.  was  exerted  at  #,  what 
force  was  exerted  at  a  to  cut 
the  tin? 

§67.  Additional  Problems  on  Town  Block  and  Lots.  —These  problems 
are  based  on  Fig.  3,  page  2.     Fig.  35  is  a  detail  drawing  of  the 
street  and  sidewalk  crossings  at  one  of  the 
four  corners  of  the  block. 

1.  The  city  paves  all  street  crossings. 
If  the  cost  of  material  is  $1.20  per  sq.  yd. 
and  of  labor  for  excavating  and  constructing 
is  $1.05  per  sq.  yd.,  what  will  be  the  expense 
to  the   city  of   paving    the    4   rectangular 
street  (not  sidewalk)  crossings   such    as  1 
(Fig.  35)?     Ans.  $1006.40. 

2.  Water   mains   run  along    Race   and 
Market  streets,  and  they  are  connected  by  a 
main  through  the  alley.     To  put  in  these 

mains  a  special  water  tax  of  27|#  per  foot  of   street  and  alley 


L 

j 

*~c^~~* 

\\V>\\\\\\\\\ 

ffl 

1 

Race 

I 

5J?'    ^. 

s 

UJ 

FIGURE  35 

frontage  was  assessed  by  the  city  against  all  property  abutting  on 


92  RATIONAL   GRAMMAR   SCHOOL   ARITHMETIC 

the  mains.    What  tax  must  the  owner  of  lot  A  pay?  of  lot  B? 
1st.  Ans.  $27.50. 

3.  Make  and  solve  problems  similar  to  problem  2  for  other  lots. 

4.  Each  property   holder  is  required  to  pay  for  the  material 
for  a  concrete  sidewalk  in  front  of  his  premises.     The  walk  is  to 
be  6  ft.  wide,  and  the  material  costs  27^  per  square  foot.     Omit- 
ting the  corner   squares   (such  as  2,  3,  4,  5,  Fig.   35),  find  the 
expense  to  each  lot  in  the  block  for  the  sidewalks. 

5.  If   the   owner   of    lot    L   pays   for   the   material   for   the 
square  of  sidewalk  (2,  Fig.  35)  at  his  corner  of  the  block,  how 
much  will  this  increase  his  'assessment? 

6.  The  cost  for  labor   in   making  the   sidewalks  is   13^   per 
square   foot.     What  will  be  the  entire  expense  for  both  labor  and 
material  for  the  sidewalks  of  the  whole  block?     (Omit  strips  such 
as  6,  7,  8  and  9.) 

7.  An  electric  lamp  is  placed  at  each  of  2  opposite  corners  of 
the  block  and  a  gas  lamp  at  each  of  the  other  2  corners.     The  cost 
of  maintaining  a  gas  light  is  $36  a  year;  an  electric  light,  $172 
a  year.    The  entire  cost  of  the  2  gas  lights  and  '£  of  the  cost  of  the 
2  electric  lights  is  assessed  against  the  property  holders  of   this 
block.     What  will  be  the  yearly  assessment  against  the  whole  block 
to  meet  the  expenses  of  lighting? 

8.  Each   property  holder    pays   in   proportion   to    his   street 
frontage.      Make   problems   on   the   cost   to  individual   lots  for 
lighting  the  streets  around  this  block. 

§68.  Additional  Problems  on  House  Plans. — These  problems  refer 

to  Fig.  6,  page  7. 

1.  Masons  count  22-j-  bricks  per  cubic 
foot  for  chimney  and  solid  work.  The 
cross-section  of  the  kitchen  chimney  is  a 
rectangle  2  ft.  by  4  ft.  (Fig.  36)  and  the 
chimney  is  42  ft.  tall.  Allowing  ^  of  the 
number  of  cubic  feet  in  the  chimney  for 

PIGUBB36  the  twQ  flueg  runniDg  from  the  bottom  to         FIGURE  w 

the  top,  how  many  bricks  are  needed  to   build   it?     Ans.   7200. 
2.  The  cross-section  of  the  fireplace  in  the  hall  is  half  of  a 


DIVISION  93 

6-ft.  square  (Fig.  37)  to  a  height  of  26  ft.  from  the  basement 
to  the  garret  floor,  where  it  joins  the  kitchen  chimney.  Allowing 
-Jg  of  the  number  of  cubic  feet  in  the  fireplace  for  the  flue  and 
the  2  fireplace  openings,  how  many  bricks  will  be  needed  for  the 
fireplace?  Am.  9945. 

3.  What  is  the    cost  of  hauling   the  brick  for  foundations, 
chimneys,    and   porch    piers,    at     $1.50    per   thousand?*      (See 
problems  11  and  14,  p.  6.) 

4.  The  walls  and  ceilings  of  all  downstairs  rooms  are  to  be 
tinted  at  20^  per  square  yard,  no  deductions  of  any  kind  being 
made.     The  left  side  wall  of  the  dining-room  is  to  be  considered 
as  straight.     The  walls  downstairs  are  9  ft.  high.     Find  the  cost 
of  tinting. 

5.  The  kitchen  floor  is  to  be  covered  with  linoleum,  which 
comes  in  1^-yd.  widths,  at   $1.12£  per  yard  of  length.     What 
will  it  cost? 

6.  What  will    it  cost    to   carpet  all  the    bedrooms    and  the 
upstairs  hall  (not  including  the  landing  at  the  head  of  the  stairs) 
with  ingrain  carpet  1  yd.  wide,  at  55^  per  yard  of  length?     In 
rooms  not  square,  strips  are  to  run  the  long  way  of  the  room. 

7.  What  will  it  cost  to  cover  the  bathroom  floor  with  tiles  at 
45^  per  square  foot  laid? 

8.  The  basement  walls  are  8  ft.  high  and  the  arrangement  of 
the  rooms  is  the  same  as  on  the  first  floor  plan.     What  will  it  cost 
to  cement  the  floor  and  the  inside  walls  of  all  basement  rooms  at 
190  per  square  yard,  making  no  allowance  for  openings? 

9.  What  will  be  the  cost,  if  allowance  is  made  for  the  open- 
ings given  in  problems  9  and  10,  p.  6? 

10.  The  4  outside  walls  of  a  brick  store  building   are  to  be 
3  brick   walls,   to   extend  to  a  depth  of  6'  and   to  a  height  of 
36  feet.     Two  of  the  walls  are  to  be  40'  long  and  2  are  to  be  25' 
long.     There  are  to  be  also  3  inside  cross-walls   each  2  bricks 
thick  and  25'  long.     What  will  the  brick  for  all  these  walls  cost 
at   $9.50   per   thousand.       (Use   the   nearest   thousand).      Ans. 
$2410.50. 

*  See  second  footnote,  p.  6. 


94 


RATIONAL    GRAMMAR    SCHOOL    ARITHMETIC 


BILLS  AND  ACCOUNTS 


568.  Exercises. 


MON. 

Tu. 

WED. 

TH. 

FRI. 

SAT. 

WAGES 

HOURS 

HOURS 

HOURS 

HOURS 

HOURS 

HOURS 

$ 

CTS. 

Adams  

iy* 

8 

8 

sy* 

9 

8^ 

Benson 

8 

8K 

SX 

8£ 

8K 

9 

Boyd  

o 

4 

6 

6 

6* 

7 

Claussen  

8 

8X 

8% 

8K 

9 

8M 

Denning  

8K 

9 

8^ 

8K 

9 

9 



Doan  

8 

8 

8 

8 

8 

8 

1.  Find  the  number  of  hours  each  man  worked. 

2.  Find  the  number  of  hours  all  the  men  worked. 

3.  At  30<p  an  hour,  find  the  daily  wages  of  each  man. 

4.  Find  the  amount  due  each  man  at  the  close  of  the  week. 

5.  Find  the  amount  due  to  the  six  men  at  the  close  of  the  week. 

6.  "Which   man  worked   the   greatest   number   of   hours?  the 
smallest  number? 

7.  Counting  9  hr.  to   the  day,  what  is   the  greatest  amount 
any  one  man  could  earn  in  a  week? 

8.  Counting  8  hr.  to  the  day,  how  much  "overtime"  did  each 
man  work? 

9.  For  how  many  extra  hours  did  five  of  the  men  work,  count- 
ing 8  hours  to  the  day? 

10.  If  double  wages  were  paid  for  "overtime,"  how  much  did 
each  man  earn  during  the  week?     How  much  did  all  earn? 

11.  How  many  hours  did  the  third  man  lose? 


§70. 


ORIGINAL    PROBLEMS 


1.  Adams  paid  $4  on  a  doctor's  hill,  and  put  aside  $2  for  rent. 

2.  Benson   has  a  wife  and  two   children.      Average   amount 
available  for  each  of  the  family. 


BILLS   ASTD    ACCOUNTS 


95 


3.  How  much  does  Boyd's  loss  of  time  reduce  his  wages  for 
the  week? 

4.  Claussen  paid  $10  for  clothing,  and  $3.50  for  the  week's 
board. 

5.  Denning  saves  $3  a  week  to  pay  the  premium  on  a  life  insur- 
ance policy,  pays  $3.50  a  week  for  board,  and  35^  for  laundry. 

6.  Plan  Doan's  expenses. 

§71.     Account  Entries. 

A  man  has  on  hand  at  the  beginning  of  the  month  of  February, 
1902,  $250.  He  receives  during  the  month  $200  for  salary  and 
$60  for  the  rent  of  two  houses.  During  the  month  he  pays  $72.50 
on  an  insurance  policy,  $110  for  household  expenses,  $25.50  for 
incidentals,  and  $5  for  books  and  stationery.  His  account  book 
shows  the  following  entries : 
Dr.  (Receipts)  CASH  (Expenditures)  Or. 


1902 
Feb.  1 

11  15 
"27 

"28 

On  Hand 

$250 
200 
60 

00 
00 
00 

1902 
Feb.  1 
"  28 

By  Insurance  
"  Household  Exp. 
"    Incidental      " 
"  Books  and  Stat'y 

'  '  Balance 

$  72 
110 
25 
5 

297 

50 
00 
50 
00 

00 

To  Salary  ;       

"  Rent      .    .   . 

On  Hand 

/ 

510 

00 

510 

00 

297 

00 

1.  With  what  items  is  cash  the  man's  debtor?   With  what  items 
is  cash  his  creditor? 

2.  During  the  month,  how  much  money  did  the'man  receive? 
How   much  money  did  he  actually  possess  between  February    1 
and  28? 

3.  How  much  was  spent?  What  is  the  difference  between  receipts 
and  expenditures?     How  much  more  was  spent  than  was  received 
as  salary? 

Draw  forms  and  prepare  cash  accounts  for  the  following: 

4.  March,  1902.     On  hand  first  of  the  month  $1000.     March 
5,  sold  land  for  $1500;  received  on  the  15th,  $150  for  rent;  and 
sold  a  team  for  $225  on  the  25th.     Bought  real  estate  on  the  8th 
for  $1000;  paid  $05.25  for  repairs  on  the  15th;  taxes  on  the  16th 


96 


RATIONAL   GRAMMAR   SCHOOL   ARITHMETIC 


amounted  to  $17.50;  household  expenses  for  the  month  amounted 
to  $100,  and  personal  expenses  to  $50. 

5.  June,  1902.      A  boy's  receipts  and  expenditures  were  as  fol- 
lows :  Received  250  for  mowing  the  lawn  on  the  1st ;  500  a  day  for 
running  errands  on^the  4th  and  5th;  250  for  mowing  the  lawn 
again  on  the  loth.     On  the  17th  he  paid  $1.25  for  a  hat.     The 
same  day  he  earned  500  for  delivering  packages  for  the  grocer. 

6.  July,  1902.      A  farmer   sold   on   the   13th  two  loads   of 
hay  at  $20;  4  hogs  weighing  in  all  1000  Ih.  at  60;  15  Ib.  butter 
at  200.     On  the  13th  he  bought  26  Ib.  granulated  sugar  at  60; 
3  Ib.  coffee  at  350;  and  dry  goods  to  the  amount  of  $15.     On  the 
26th  he  sold  vegetables  to  the  amount  of  $1.25,  and  9  doz.  eggs  at 
100.     The  same  day  he  bought  clothing  to  the  amount  of  $10. 

7.  Find  cost  of  each  purchase,  also  the  amount  due,  on  the  bill 
below : 

E.  S.  WICKWIRE, 

In  account  with  MANNHEIMER  BROS.,  Dr. 


1902 
Jan    4 

To  1  Rug                    at  $75  50 

"     4 

'  '  1  Overcoat       "     35  00 

"    12 
"    14 

"  7yd.  Dress  Goods  ....    "       2.00 
"  3%  "  Silk  "       150 

Amount  due                              





MANNHEIMER  BROS. 


Received  payment, 
Jan.  31,  1902. 

Draw  forms  and  prepare  similar  bills : 

8.  Edward  Morris  in  account  with  J.  W.  Bowlby,  Dr.,  St.  Paul, 
Minn.,  May  30,  1902.     1  doz.  collars  at  250  a  piece;  |  doz.  pr. 
cuffs  at  25^  a  pair;  2  neckties  at  500;  3  shirts  at  $1.25;  1  hat  at 
$3.50;  1  pr.  shoes  at  $3.50;  1  pr.  gloves  at  $1.25.     Paid,  May  30. 

9.  A.   H.   Simons  bought  of  Yerxa  Bros.,  Chicago,  June   6, 
1902,  1  sack  appleblossom  flour  at  $1.30;  8  Ib.  ham  at  170;  15 
Ib.  granulated  sugar  at  60;  |  Ib.  oolong  tea  at  $1.60;  3  Ib.  Mocha 
and  Java  coffee  at  400 ;  1  bunch  of  celery  at  300.     Paid  in  full  the 
same  day. 


BILLS   AND    ACCOUNTS 


97 


10.  Edward  Ryan  bought  of  M.  J.  Doran,  February  7,  1902, 
2  cords  of  hard  wood  at  $8.75;  3  T.  of  hard  coal  at   $8.25;  2 
cd.  of  pine  slabs  at  $3.25. 

11.  May  4,  1902,  Alvin  Johnson  bought  of  the  John  Martin 
Lumber  Co.,  cash  payment,  cedar  shingles  at  $3  per  M;  1  bbl.  of 
lime  at  G8# ;  1000  ft.  of  pine  flooring  at  55^  per  M ;  3000  laths 
at  $1.50;  15  Ib.   shingle  nails  at  7#. 

Accounts  are  not  always  settled  in  full  at  the  close  of  the 
month.  When  a  portion  of  the  amount  due  is  carried  over  to  the 
next  month,  this  item  appears  as  the  first  item  on  the  bill,  under 
the  title  "accounts  rendered." 

12.  Find  amount  of  this  bill : 


J.  FIRESTONE. 


To  ALLEN,  MOON  &  Co.,  Dr. 


1902 
Feb.  1 

To  Acc't  rendered                              .  • 

24 

85 

"    10 

"  2  Sacks  of  Flour               at  $2  00 

"    10 

«  25  Ib  Ham                                    18 

"    10 

'  '   4  doz.  Eggs  22 

"    10 

'  '   6  Ib   Butter                          .         25 

"    15 
"    15 

"  25  Ib.  Granulated  Sugar.  .  .      .06 
"  Vegetables  .  ._  5.00 

By  Cash  

13.  A.  L.  Parker,  in  account  with  Marshall  Field,  lacks  $12.75 
of  paying  the  entire  bill  of  February,  1902.     His  March  purchases 
were  as  follows:    March  5,  J  doz.  towels,  @  500  a  piece;  2  table 
cloths,  @  $0.50;  1  doz.  napkins,  @  $5.50  a  doz.;    2  bed  spreads 
@  $2.50;  |  doz.  sheets  @  75<^  a  piece;  J  doz.  pillow  cases  @  20^ 
a  piece;  7  yd.  dress  goods  @  $1.50;  findings,  $5.25.    Paid  in  full 
March  28. 

14.  Edward  D.  Young  in  account  with  F.  W.  Salisbury,  wood 
and  coal  dealer,  has  left  over  from  a  previous  account  $8.25.     On 
the  3d  of  January,  1902,  he  bought  2  T.  of  hard  coal  @  $8.25;  1 
cd.  of  hard  maple  wood,  $8.50;  1  cd.  of  pine  slabs,  $3.50.     Paid 
the  whole  amount  January  29, 


98 


RATIONAL   GRAMMAE    SCHOOL   ARITHMETIC 


15.  On  Nov.  15,  I  purchased  1  can  lima  beans  @  120;  1  can 
peas  @  12^;  1  qt.  cranberries  @  120;  4  Ib.  pork  chops  @  120; 
4|  Ib.  chicken  @  120 ;  2  Ib.  butter  @  320,  and  paid  the  account 
with  a  $5.00  bill.  What  change  was  due  me? 

§72.  Problems  of  the  Grocery  Clerk. — A  boy  clerk  in  a  grocery 
store  made  the  following  transactions  including  the  filling  of 
the  orders,  and  returned  the  correct  change  without  an  error  all 
in  one  hour.  What  change  did  he  return  to  each  customer? 

1.  The  first  customer  bought 

Celery $0.05         Potatoes $0.15 

Eggs 22         Sugar.  .  10  Ib.  @  4J0. 

and  paid  the  clerk  with  a  $1.00  bill. 

2.  The  second  customer  bought 


Salt $0.10 

Flour 55 

Yeast 02 

Milk 07 

and  he  paid  with  $1.50. 

3.  The  third  customer  bought 

Tomatoes $0.20 

Soap 25 

Butter.. 3  Ib.  @  270. 
Coffee.. 3  Ib.  @  33^0. 

and  paid  with  a  $5.00  bill. 

4.  The  fourth  customer  bought 

Eggs $0.22 

Bacon 18 

Peas 13 

and  paid  with  a  $5.00  bill. 

5.  The  fifth  customer  bought 

Cornmeal $0.25 

Sweet  potatoes 27 

Candy 10 

Grapes 18 

and  paid  with  a  $2  bill. 


Apples $0.25 

Breakfast  food 13 

Kice..3  Ib.  @  8J0. 


Beans 4  qt.  @  70. 

Tea 2  Ib.  @  600. 

Cauliflower.  .3 heads  @  180. 


Lampwicks $0.10 

Cheese..  .16 


Melon $0.25 

Pineapple 20 

Lard.. 3  Ib. 


BILLS   AHD   ACCOUNTS 


Tea 21b.  @  550. 

Flour 1  sack,  $1.15 

Rice.  .          .  .6  Ib.  6 


6.  The  sixth  customer  was  a  farmer,  who  sold  to  the'groder^ 

Eggs 6  doz.  @  180.         Tomatoes.  .  .2  bu.  @  750. 

Butter 15  Ib.  @  220.         Sugar  corn.  .  6  doz.  @  100. 

Potatoes.  . .  .5  bu.  @  350. 
and  bought  of  the  grocer 

Sugar 20  Ib.  @  4^0. 

Coffee 31b.  @  250. 

Cheese 2  Ib.  @  180. 

How  much  money  should  the  grocer  pay  the  farmer  to  balance  the 
account? 

A  grocer's  daily  sales  for  4  weeks  are  given  below.  Without 
rewriting  the  numbers  find  the  total  sales  (a)  for  each  week ;  (b) 
the  average  for  each  day  and  (c)  the  total  for  the  4  weeks : 

MON.          TUBS.  WED.          THUB.  FBI.  SAT.  TOTALS 

$28.75     $37.25     $35.18     $68.12     $20.13     $86.58     

18.62         9.68       21.83       40.28       37.60       75.76     

30.18       29.95       23.61         9.61       10.84       68.94     

19.27       39.13       28.16       38.19         5.63       98.56     

§73.  Family  Expense  Account. 

Foot  the  following  problems  as  rapidly  as  you  can  work  accu- 
rately. 

1.  The  household  expenditures  of  a  family  are  here  given  for 
each  of  the  first  7  da.  of  Oct.  1900.  Find  the  daily  expenditures 
and  the  total  expenditure  for  the  week. 

Oct.  1.                            Oct.  2. 
Oatmeal $0.11     Laundry $0.48 


Meat 18 

Peaches 25 

Grapes 13 


Dried  beef. 

Tea 

Bread  

Celery 
Soap 


Total.. 


,10 
,30 
.20 
.05 
.25 


Sweet  potatoes 

Apples 

Bread 

Meat 

Sundries 

Bread 

Crackers 

Total.. 


Oct.  3. 

Bread $0.15 

,18     Baking  powder     .25 

.25     Cheese 10 

,10     Macaroni 15 

,20     Bananas 15 

.63     Honey 20 

.20     Peanuts 05 

.15     Tomatoes 15 

—     Shoe  polish  .  .      .10 

Carfare 10 

Onions 10 

Total.. 


100 


RATIONAL   GRAMMAR   SCHOOL   ARITHMETIC 


Oct.  4: 

Bread $0.10 

Meat 10 

Lunch  boxes. .     .26 
School  supplies     .75 

Car  fare 05 

Milk 3.00 

Apples 20 

Coffee .35 

Potatoes 15 

Beefsteak 35 

Tomatoes 10 

Servant 3.50 

Candy 05 

Total., 


Oct.  5. 

Bread $0.10 

Meat ... .     .10 

Car  fare 35 

Cakes 10 

School  supplies  4.50 

Carfare 10 

Meat 23 

Celery 05 

Tomatoes 05 

Cakes 15 

Sundries 57 

Carfare 25 

Beans 10 

Total.. 


Oct.  6. 

Bread $0.20 

Bacon 15 

Fruit 40 

Merchandise..   5.25 

Meat 42 

Eggs 20 

Theatre 1.50 

Bonbons 10 

Newspaper ...     .02 
Sundries 35 

Total 

Oct.  7. 
Newspaper  ...   0.10 


TOTALS 


Oct.  1st. 
Oct.  2d  . 
Oct.  3d  . 
Oct.  4th. 
Oct.  5th. 
Oct.  6th. 
Oct.  7th 


Total  1st.  wk.. 


2.  The  monthly  expenditures  of  this  family  for  clothing  are 
here  tabulated,  beginning  with  Oct.  1900.  Find,  without  rewrit- 
ing the  numbers,  the  total  expenditures  for  clothing  (a)  for  the 
first  six  months ;  (b)  for  the  last  6  months : 


OCT.  Nov.  DEC.  JAN. 

$25.50     $48.37     $15.78     $5.83 


FEB.  MAR. 

$20.26     $37.90 


TOTAL 


APR.  MAY  JUNE         JULY          AUG.  SEPT. 

$3.75     $18.64     $25.95     $8.47     $12.35     $20.16 


BILLS    AND    ACCOUNTS  101 


2.  The  daily  totals  for  the  expenditures  of  this  family  are  here' 
given  for  the  whole  year  beginning  "Oct.  1.  Find  the  monthly 
totals  and  the  total  for  the  entire  year. 


DATE 

i....     i 

OCT. 
>   51.57 

Nov. 

$  55.28     : 

DBG. 
$  53.15 

JAN. 
$      3.70 

FEB. 
$        .35      1 

MAR. 

$     2.86 

2  

8.94 

3.65 

.13 

51.26 

58.10 

56.55 

3 

2.09 

7.25 

1.68 

3.67 

1.02 

.60 

4 

7.35 

.25 

3.75 

.55 

5.16 

3.16 

5  
6  . 

1.50 

8.22 

1.65 

4.78 

1.85 
5.75 

5.90 
.12 

1.37 
2.15 

4.60 
6.51 

7  

.05 

2.83 

1.25 

1.60 

3.17 

2.75 

8  ..  .. 

4  75 

3.86 

8.21 

6.12 

1.16 

1.28 

9 

1  77 

2  28 

.68 

3.26 

4.18 

3.82 

10  
11  

4.13 
9.10 

6.98 
.15 

1.86 
1.17 

1.81 
3.60 

.22 

6.03 

.75 
2.29 

12  .  ... 

1.19 

3.65 

4.76 

2.31 

2.58 

1.69 

13  

3.87 

1.63 

5.16 

.05 

1.15 

9.60 

14 

10 

.85 

1.11 

5  86 

5.60 

.17 

15 

2  17 

6  87 

2  68 

1  03 

1.81 

1  25 

16  

3.98 

2.37 

.68 

4.32 

2.81 

2.18 

17  

18.47 

8.86 

3.27 

9.82 

.75 

.85 

18  .. 

1.80 

12 

7.32 

2.18 

.81 

5.63 

10 

2  55 

4.10 

1.28 

3.28 

15.61 

3.05 

20  
21  

8.81 
.25 

1.17 
1.68 

2.18 
11.12 

.75 
6.29 

5.78 
4.78 

1.06 
11.61 

22  

.25 

3.76 

16.25 

1.18 

1.05 

4.28 

213  

3.55 

9.26 

.22 

2.28 

3.27 

2  57 

24  . 

55 

7.81 

3.20 

7  60 

1.60 

.60 

25  
26  
27  

28  ,..:.. 
29  
30  
31  

1.65 
3.70 
5.74 

.76 
2.80 
3.86 

2.85 

.55 

3.86 
7.52 
.28 
18.26 
1.60 
.00 

1.68 
.54 
3.48 
6.15 

2.97 
.21 
2.08 

3.91 
2.83 
.25 

1.65 
5.16 
18.21 
6.38 

2.86 
8.07 
3.35 
4.15 
.00 
.00 
.00 

3.05 
3.18 
1.75 
3.10 
6.21 
.76 
.15 

Monthly  > 
Totals   f 

3.  Find  the  daily  average  per  month  for  each  of  the  6  months. 

4.  Find  the  monthly  average  for  the  6  months  from  October 
to  March. 

5.  Add  horizontally  and  find  the   total  expenditure  for  the 
first  day  of  the  6  months;  for  the  second  day. 


roa 


'  RATIONAL    GRAMMAR    SCHOOL    ARITHMETIC 


1 
2 
3 

4 
5 
6 

7 
8 
9 

10 

11 

13 

14 

15 

16 

17 

18 

19 

20, 

21 

22 

23 

24 

25 

26, 

27, 

28. 

29, 

30, 

31 


Monthly  } 
Totals    j 


52.86 
3.87 
3.68 
6.36 
1.86 
2.15 

.75 
1.95 
3.28 
2.18 
4.10 
2.15 
1.00 

.10 
1.61 
3.21 

.57 
3.76 

.15 
4.25 

.86 

.95 

.87 
1.16 
3.15 
1.82 
2.78 

.60 
3.25 

.96 

.00 


MAT 

JUNE 

JULY 

AUG. 

SEP 

53.48 

$  56.75 

$  56.73 

$  55.10 

$       .65 

2.50 

.67 

1.28 

2.50 

54.10 

2.18 

1.20 

2.10 

6.82 

4.06 

1.28 

.85 

3.50 

.04 

.60 

.25 

.75 

4.05 

.00 

2.75 

8.12 

3.80 

1.10 

1.28 

.25 

3.62 

2.37 

.05 

.75 

1.52 

.67 

2.12 

.12 

4.25 

.68 

4.10 

.55 

.15 

1.18 

2.12 

.10 

.96 

.35 

2.60 

.39 

1.01 

2.84 

3.60 

.00 

1.86 

.86 

3.69 

.10 

3.10 

3.24 

1.52 

2.50 

2.18 

.69 

3.27 

.78 

.55 

.00 

1.58 

1.68 

2.58 

1.86 

1.57 

3.82 

.55 

3.65 

1.68 

.76 

.55 

1.68 

.25 

2.75 

.56 

6.53 

2.17 

2.51 

.99 

3.65 

.28 

.69 

.86 

6.24 

.05 

.12 

3.00 

1.76 

3.26 

8.29 

.17 

.12 

2.08 

.25 

.00 

2.16 

6.81 

1.15 

4.86 

.00 

2.60 

.05 

3.12 

.28 

1.68 

.05 

.10 

1.28 

.83 

1.00 

8.16 

7.06 

3.75 

1.96 

2.28 

.15 

.19 

.15 

1.76 

.05 

.08 

2.60 

1.67 

3.75 

6.28 

1.20 

2.97 

2.18 

1.28 

.00 

2.00 

3.60 

1.01 

8.14 

3.16 

3.60 

.10 

3.87 

1.00 

2.15 

.25 

2.65 

.75 

.00 

1.58 

1.68 

.00 

TOTAL 


6.  Find  the  daily  average  per  month  for  each  of  the  6  months. 

7.  Find  the   monthly  average   for   6   months  from  April  to 
September. 

8.  Find  the  monthly  average  for  the  whole  year. 

9.  Find  by  adding  horizontally  the  total  expenditure  for  the 
first  day  of  these  6  months  ;  for  the  second  day. 


BILLS    AND    ACCOUNTS  103 

§74.  The  Equation. 

1.  y  Ib.  of  sugar  are  placed  on  the  left  scale  pan,  and  a  weight 
of   10   Ib.    on   the   right  pan   balances   it    (see   Fig.   38).     How 
heavy  is  *•? 

The  balance  of  these  two  weights 
is  expressed  thus :  y  =  10.  (I)  ^T^^Q^b  10 Lb 

This  expression  is  called  an  equa- 
tion, and  is  read  ily  equals  10." 

2.  If  a  4  Ib.  weight  is  now  added 
to  the  right  pan,  how  many  additional 
pounds  of  sugar  must  be  placed  upon 

the  left   pan   to   balance  the  scales?  FIGURE  38 

Write  an  equation  to  express  the  relation  between  the  weights 

now  in  the  pans. 

3.  If  x  Ib.  on  the  right  balance  y  Ib.  on  the  left,  how  would 
you  state  the  fact  in  an  equation? 

4.  If  20  Ib.  are  added  to  the  x  Ib.  already  on  the  right,  how 
many  pounds  must  be  added  on  the  left  to  balance  the  equation? 

5.  If  35  is  added  on  the    right    side    of    equation  (I),  what 
change  must  be  made  on  the  left  side  to  balance  the  equation? 
Write  the  equation  thus  changed. 

Just  as  the  horizontal  position  of  the  scale  beam  shows  that 
there  is  a  balance  of  the  weights  on  the  pans,  so  the  equality  sign, 
=  ,  shows  that  there  is  a  balance  of  value  of  the  numbers  between 
which  it  stands.  It  must  never  be  used  between  two  numbers  that 
do  not  balance  in  value. 

The  number  on  the  left  of  the  sign  of  equality  is  called  ihe  first 
member,  or  the  left  side  of  the  equation.  The  number  on  the  right 
is  called  the  second  member,  or  the  right  side. 

6.  Four  equal  weights  and  a  5  Ib.  weight  just  balance  21  Ib. 
(Fig.  38.).     How  heavy  is  one  of  the  equal  weights? 

SOLUTION. —    4#4-5  =  21.     Subtract  5  from  both  sides 
15    -5 


4#    =16.     If  4x  =  16  what  is  xt   What  is  the  answer 
to  the  problem? 

7.  3#+2  =  14;  findy.      8.   5z  +  12  =  32;  find  x.      9.  7z  +  8  = 
29;  find  x.     10.  9z  +  3  =  75;  find  z.     11.   15a  +  3  =  48;  find  a. 


104  RATIONAL   GRAMMAR    SCHOOL   ARITHMETIC 

CONSTRUCTIVE  GEOMETRY 
§75.  Problems  with  Ruler  and  Compass. 

Problems  I.  to  XL  are  to  be  solved  with  ruler 
and  compass.  Keep  all  the  pencil  points  sharp 
while  drawing  and  work  carefully. 

The  simple  instrument  shown  in  Fig.  39  is  a 
form  of  the  compass  which  will  do  for  these 
problems. 

The  pencil  point  of  the  compass  will  be 
called  the  pencil  foot,  or  pen  foot.  The  other 
point  will  be  called  the  pin  foot. 

PROBLEM    I.  —  Draw    a   circle   with    \   inch 
FIGURE  39     '    radius. 

EXPLANATION.  —  First  Step  :  Place  the  pin  foot  on  an  inch  mark  of  your 
foot  rule  and  spread  the  compass  feet  apart  until  the  pencil  foot  just 
reaches  to  the  next  half  inch  mark. 

Second  Step:  Without  changing  the  distance 
between  the  compass  feet,  put  the  pin  foot  down  at 
some  point,  as  A,  Fig.  40,  of  your  paper  and,  with 
the  pencil  foot,  draw  a  curve  entirely  round  the 
point  A. 

A  curve  drawn  in  this  manner  is  called  a  circle. 
How  far  is  it  from  A  to  any  point  of  the  curve? 

Any  part  of  the  whole  circle,  as  the  part  from  C 
to  D,  or  from  D  to  B  is  an  arc  of  the  circle. 

The  point  A,  where  the  pin  foot  stood,  is  the  PIGUKE  40 

center. 

The  distance  straight  across  from  B,  through  A  to  C,  is  a  diameter. 
BC  is  a  diameter. 

The  distance  from  A  straight  out  to  the  circle  is  a  radius.  The 
plural  is  radii  (ra'-di-I).  AC,  AB  and  AD  are  all  radii. 

What  part  of  the  diameter  equals  the  radius? 

EXERCISES 
1.  Draw  circles  with  these  radii: 


2.  Draw  circles  with  the  same  center  A,  and  with  these  radii  : 

i";  i";  i";  i;  if- 

Circles  whose  centers  are  all  at  the  same  point  are  called  con- 
centric circles. 

PROBLEM  II.  —  Draw  a  line  equal  to  a  given  line. 


CONSTRUCTIVE    GEOMETRY  105 

EXPLANATION.— Let  the  given  line  AB  (Fig.  41)  have  the  length  a 
units. 

With  ruler  and  pencil  draw  any  straight  line,  as  CX,  longer  than  a. 

Place  the  pin  foot  on  A  and  spread  the 

compass  feet  until    the  pencil    foot    just    ^ a -g 

reaches  to  B.    Without  changing  the  dis-  \j) 

tance  between  tne  feet,  put  the  pin  foot  on    £  ~  X 

C  and  with  the  pencil   foot  draw  a  short  FIGURE  41 

arc  across  the  line  CX  as  at  D. 

Then  CD  is  the  desired  line;  for  if  we  call  its  length  x,  we  have  made 
x  =  a. 

EXERCISES 

1.  Draw  lines  equal  in  length  to  these  given  lines: 
a  Z>  c 


2.  Draw  lines  having  these  lengths: 

1".     11".     Ol".     O3".     QT" 

1  .;  If  ;  4f  ,;  2f  ;  3$  . 

PROBLEM  III. — Draw  a  line  equal  to  the  sum  of  two  or 
more  given  lines.  , 

EXPLANATION. — Let  the  two  given  lines  be  a  and  6,  Fig.  42.  First  step : 
Draw  the  indefinite  line  CX  longer  than  the  combined  length  of  a  and  b, 

and  make  CD  equal  to  a  as  in 

? Problem  II. 

b  Second  Step:  Spread  the  com- 

pass feet  apart  as  far  as  the  length 

v     of  6.     Then  put  the  pin  foot  on 

the  crossing  point  (intersection) 
of  the  arc  and  line  at  D  and  draw 
another  short  arc  at  E,    How  long 
is  DEI    How  long  is  CEt 
CE  is  the  desired  sum.     If  we  call  its  length  s,  we  may  write  this 
equation : 

(I.)    s  =  a  +  b. 

With  ruler  and  compass  how  can  you  find  a  line  equal  to  the  sum  of 
3  given  lines?  of  4?  of  any  number  of  given  lines?  This  is  called  con- 
structing the  sum  of  lines. 

EXERCISES 

1.  Construct  the  sum  of  the  lines  in  each  column,  denote  the 
constructed  line  by  s,  and  write  an  equation  like  (I.)  for  each 
case: 

cam  a 

den  b 

e  p  m 


106  RATIONAL   GRAMMAR    SCHOOL   ARITHMETIC 

2.   Construct  these  sums : 

PROBLEM  IV. — Draw  a  line  equal   to  the  difference  of   two 
given  lines. 

EXPLANATION. — Let  the  two  given  lines  be  a  and  b.  First  step:    Draw 
the  indefinite  line  CX  longer  than  the  minuend  line  a.     As  above,  make 

the  minuend  CD,  equal 
• '  ••  to  a. 

b Second  Step :  Spread 

~  the  compass  points  apart, 

U  T7"  QG      V»£*fYiV£i       OC       "f  tJ  V      Q  C!       t  1  i  i_i 


C__J 


as  before,  as  far  as  the 
length    of     the    subtra- 


FIGTJBE  43  hend    line    b.      This    is 

called  "taking   b    as   a 

radius."  Place  the  pin  foot  on  D,  swing  the  pencil  foot  back  toward 
C,  and  draw  the  arc  at  E  across  the  line  CX.  This  makes  DE  how 
long?  What  line  now  equals  the  difference  d  between  a  and  6?  The 
equation  for  this  case  is : 

(II.)    d  =  a-b. 

EXERCISE 

Construct  the  differences  of  these  pairs  of  lines,  call  each  dif- 
ference dy  point  it  out  in  the  construction  and  write  the  equation 
for  each  case : 

m c  f  a 


PROBLEM  V.  —  Draw  a  line  equal  to  2,  3,  or  4  times  a  given  line. 
EXPLANATION.  —  Let  a  be  the  given  line.     We  are  simply  to  draw 


a  line  equal  to  the  sum 

of  a  and  a.     (See  Prob-        _  a 

lemlll.) 


D  E 


We  may  write  r 

(III.)    p  =  2a. 

How  would  you  con-  P 

struct   3a?    4a?    5a?   7a?  FIGUBE  44 

oxi  &y{  vzc 

Write  and  read  the  equation  for  each  construction. 

EXERCISE 

Construct   three   times  these   lines  and   give  their  equations 
(like  III.,  above): 

a  c  d  %a  3m 

PROBLEM  VI. — Divide  a  given  line  into  two  equal  parts. 


CONSTRUCTIVE    GEOMETRY 


107 


B 


DEFINITION.— Dividing  a  line  into  two  equal  parts  is  called  bisecting 
the  line. 

EXPLANATION.— Let  the  given  line  be  AB,  Fig.  45,  call  its  length  a. 
First  step:  Spread  the  compass  feet  apart  a  little  farther  than  £  the 
length  of  AB.  Ple^?  the  pin  foot 
first  on  A  and  bring  the  pencil  / 

foot,  by  estimate,  somewhere 
above  the  middle  of  the  line  a, 
and  draw  the  arc  1.  Then  carry 
the  pencil  foot  down,  by  estimate, 
below  the  middle  of  a  and  draw 
the  arc  2.  These  arcs  should  both 
be  drawn  long  enough  to  make 
sure  that  arc  1  passes  over  arc  2, 
under  the  middle  point. 

Second  step :  Now  change  the 
pin  foot  to  B,  and  without  chang- 
ing the  distance  between  the  feet, 
draw  arc  3  cutting  arc  1  and  4 
cutting  arc  2.  Call  the  crossing 
points  C  and  D. 

Third  step :   Place  the  edge  of 
the  ruler  on  C  and  D  and  draw 
the  straight  line   CD,  crossing  a   at  E.      Either  AE  or   BE  is  equal 
to  £  of  a.     Test  by  measuring  with  the  compass. 

The  equation  for  this  case  is: 

(IV.)     q  =  \a,  or  q  =  j. 

The  first  is  read  "q  equals  one-half  a,"  meaning  q  equals  \  of  a,  and  the 
second  is  read  "q  equals  a  divided  by  2."  Do  they,  therefore,  mean 
the  same  thing? 


FIGURE  45 


EXERCISES 


1.  Draw  a  line  f"  long  and  bisect  it. 

2.  Bisect  lines  of  these  lengths: 

J";    If;    3";    5";    5i";    3J". 


FIGURE  46  FIGURE  47 

3.  Longer  lines  may  be  bisected  at  the  blackboard  with  crayon 
and  string  (Fig.  40) ,  or  on  the  ground  with  a  cord  and  a  sharp 
stake  (Fig.  47). 


108  RATIONAL    GRAMMAR    SCHOOL   ARITHMETIC 

PROBLEM  VII. — Construct   an   equilateral    (equal  sided)  tri- 
angle with  each  side  equal  to  a  given  line. 

EXPLANATION.— Let  a  Fig.  48  be  the  given  line. 

Draw   CX  longer  than  a  and 

make  CD  =  a,  as  in  Problem  II. 

With  length  a  between  the 
compass  feet,  place  the  pin  foot  on 
C  and  draw  arc  1,  by  estimate, 
above  the  middle  of  CD. 

Without  changing  the  distance 
between  the  compass  feet,  place 
the  pin  foot  on  D  and  with  the 
pencil  foot  cut  arc  1  with  arc  2. 
Call  B  the  intersection  (crossing 
FIGURE  48  point)  of  arc  1  and  arc  2.  With  the 

ruler  draw  a  line  from  C  to  B  and  another  from  D  to  B.     Then  CDB  is 
the  desired  equilateral  triangle. 

EXERCISES 

1.  Construct  equilateral  triangles  having  sides  of  these  lengths: 
bed  e 


2.  Construct  equilateral  triangles  having  these  sides : 

1";    2";    3";    2J";    4". 

3.  With  crayon  and  string  construct  these  equilateral  triangles 
on  the  blackboard : 

6";    1';    14";    18";    24". 

4.  With   cord  and  nail  or   stake  construct  these   equilateral 
triangles  on  the  floor  or  ground : 

6';    10';    18';    1  rod;    30'. 

PROBLEM  VIII. — Construct  an  isosceles  (i-sos'-see-lees)  trian- 
gle with  the  base  and  the  two  equal  sides  equal  to  given  lines. 

DEFINITION.  —  An    isosceles  tri-  ^ 

angle  has  at  least  two  sides  equal. 

EXPLANATION. — Let    b,    Fig.   49,       . £ 

equal  the  base,    and  e,   one  of  the 
equal  sides. 

Make  CD  =  b  (Problem  II).  With 
C  as  center  and  radius  equal  to  e 
draw  arc  1.  With  the  same  radius 
and  with  D  as  center  draw  arc  2. 
Connect  their  intersection  E  with  C 
and  with  D. 

Then  CDE  is    the  desired  isos- 

celes  triangle ;  for  we  made  CD  =  6,  FIGURE  49 

CE  =  e  and  DE  =  b. 

DEFINITION.— The  side,  CD,  which  is  not  equal  to  either  of  the  other 
two  sides  is  the  base. 


CONSTRUCTIVE    GEOMETRY 


109 


EXERCISES 


1.  The  figures  in  Fig.  50  represent  all  forms  of  the  triangle. 
What  is  a  triangle? 


FIGURE  50 

2.  What  triangle  has  all  its  sides  equal?     What  is  an  equi- 
lateral triangle? 

3.  What  triangles  have  at  least  2  sides  equal?      What  is  an 
isosceles  triangle? 

4.  What  triangles  have  no  two  sides  equal?     What  is  a  sca- 
lene (ska-leen')  triangle? 

PROBLEM    IX. — Draw    a   scalene    triangle   with    sides    equal 
to  given  lines    (no  two  being  equal). 


EXPLANATION. — On  the  line  CX 
make  CD  =  a  as  in  Problem  I. 
With  C  as  center  and  b  as  radius, 
draw  arc  1.  Then  with  D  as  center 
and  with  c  as  radius,  draw  arc  2 
across  arc  1.  Call  their  inter- 
section E. 

With  the  ruler  draw  line  EC 
and  ED. 

Then  CDE  is  a  scalene  triangle    (J 
with    the  sides  equal  in  length  to 
a,  fcandc.  FIGURE  51 

EXERCISES 

1.  Draw  a  scalene  triangle  having  sides  of  these  lengths: 

a  b c 

2.  Draw  a  scalene  triangle  having  sides  of  3",  4",  and  5"j 
of  1",  1J",  and  2". 

PROBLEM  X. — With  the  compass  draw  a  three-lobed   figure 
inside  of  a  circle  of  I"  radius. 


110 


RATIONAL   GRAMMAR   SCHOOL   ARITHMETIC 


EXPLANATION.— Draw  a  circle  with  £"  radius  around  some  point,  as  O, 
as  a  center.  Set  the  pin  foot  at  any  point 
on  the  circle,  as  at  1,  Fig.  52,  and  with- 
out changing  the  distance  between  the 
compass  feet  draw  the  arc  from  A  on  the 
circle  through  O  to  B  on  the  circle. 

With  same  radius  and  with  pin  foot 
on  B,  draw  a  short  arc  at  2. 

Put  the  pin  foot  on  2  and  with  the 
same  radius  as  before  draw  the  arc  BOG, 
C  being  on  the  circle  first  drawn. 

Put  the  pin  foot  on  C  and  draw  a  short 
arc  at  3.  Then  place  the  pin  foot  on  3  and 
draw  an  arc  from  C  through  O  to  A. 

FIGURE  52 


EXERCISES 


1.  Draw  a  three-lobed  figure  using  a  radius  of  1". 

three-lobed  figure  using  as  radius  this  line 


FIGURE  53 


3.  Making    the    distance    between 
the  compass  points  \"  ,  and  using  the 
points  1,  2,  3,  4,  5,  and  6  in  turn,  draw 
a  six-lobed  figure  like  Fig.  53. 

4.  Color  the  lobes  of  your  figure 
with  a  red  lead  pencil  or  with  water 
colors  and  the  spaces  a,  b,  c,  d,  e,  and 
/  with   a   green   or   yellow   pencil   or 
water  colors. 

PROBLEM  XI.  —  Draw  a  regular  C-sided  figure  (regular  hexagon) 
within  a  circle  of  \"  radius. 

EXPLANATION.  —  Draw  a  circle  with 
center  at  O  and  with  |"  radius. 

Starting  at  any  point  on  the  circle  as 
at  1,  put  the  pin  foot  on  1  and  draw  the 
short  arc  2  across  the  circle,  keeping  the 
radius  |".  Then  put  the  pin  foot  on  2  and 
draw  the  arc  3.  Then  with  pin  foot  on  3 
draw  arc  4  ;  and  so  on  around. 

How  many  steps  do  you  find  reach 
once  round? 

Now  with  the  ruler  and  pencil  connect 
1  with  2,  then  2  with  3,  3  with  4,  and 
finally  6  with  1. 

The  figure  made  by  the  G  straight  lines 
is  the  regular  hexagon  desired. 


FIGURE  54 


MEASUREMENT  111 

EXERCISES 

1.  Draw  regular  hexagons  within  circles  of  these  radii :  1" ;  2" ; 

J"  •    9J." 
,  &\  . 

2.  How  long  is  one  side  of  each  of  the  hexagons  drawn  in 
exercise  1?     How  long  is  the  sum  of  all  the  bounding  lines  of 
each  hexagon? 

DEFINITION.  — The  sum  of  all  the  bounding  lines  of  any  figure  is  called 
the  perimeter  of  the  figure. 


MEASUREMENT 
§76.  Measuring  Talue. 

Money  is  the  common  measure  of  the  value  of  all  articles  that 
are  bought  and  sold. 

The  unit  on  which  United  States  money  is  based  is  the 
dollar.  The  dollar  sign  is  $;  thus  5  dollars  is  written  $5  or  $5.00. 
This  unit  is  called  the  U.  S.  Standard  of  value. 

1.  A  dollar  is  worth  as  much  as  how  many  dimes?  nickels? 
quarters?  cents? 

To  measure  the  value  of  one  amount  of  money  by  another  is 
to  find  how  many  times  one  of  the  amounts  is  as  large  as  the 
other. 


2.  Measure  $1  by  50^;  by  25^;  by  20r/;  by  40^;  by 

3.  Measure  $10.50  by  50^;  by  75^;  by  $1.50;  by  $5.25. 

4.  A  farm  is  worth  $1000  and  a  city  lot  $2000.     Measure  the 
value  of  the  farm  by  the  value  of  the  lot. 

5.  Measure  the  value  of  a  $150  horse  by  the  value  of  a  $15  pig. 

6.  Measure  $75  by  $5;  by  $25;  by  $15;  by  $150;  by  $225. 

7.  Measure  the  value  of  160  A.  of  land  worth  $75  per  acre 
by  $100;  by  $500;  by  $1000;  by  $8000;  by  $24,000. 

8.  Which  is  worth  the  more,  80  A.  of  land  @  $75,  or  50  A. 
@  $112?  how  much  more? 

9.  Measure  $400  by  the  value  of  an  $8  calf;  of  a  $25  colt. 


112  RATIONAL   GRAMMAR    SCHOOL    ARITHMETIC 

§77.  Measuring  Length  and  Distance. 

Answer  these  problems,  first,  by  estimate,  recording  your  esti- 
mates in  a  notebook;  then  answer  by  actual  measurement. 
Finally,  compare  your  estimates  with  the  results  of  your  measure- 
ments. 

1.  How  wide  is  the  page  of  this  book?  how  long?    How  wide 
are  the  margins? 

2.  How  wide  is  a  pane  of  glass  in  your  schoolroom  window? 
how  long  is  it? 

3.  How   wide   is    the  top  of   your    desk?     how  long?      How 
high  is  the  top  of  your  desk  from  the  floor?     How  high  is  your 
seat? 

4.  How  high  is  the  bottom  of  the  blackboard  from  the  floor? 
How  wide  is  your  blackboard?  how  long  is  it? 

5.  How  tall  are  you?     How  far  can  you  reach  by  stretching 
both  arms  as  far  apart  as  possible? 

6.  How  far  is  it  around  your  waist?     How  far  is  it  around 
your  chest,  just  below  your  arms,  when  as  much   of  the  air  as 
possible    is    exhaled    from    your    lungs?      What   is    your    chest 
measurement  when  as  much  air  as  possible  is  drawn  into  your 
lungs? 

7.  The  difference  between  these  two  chest  measures  is  called 
your  chest   expansion.     What   is    your   chest   expansion?     How 
does  your  chest   expansion   compare   with  the   average   for   the 
pupils  of  your  room? 

NOTE. — You  may  easily  increase  your  chest  expansion  by  a  little 
practice  in  deep  breathing. 

8.  How  many  steps  wide  is  your  schoolroom?  how  many  steps 
long? 

9.  How  many  feet  long  is  your  room?   how  many  feet  wide? 

10.  How  many  yards  long  is  it?  how  many  yards  wide? 

11.  How  many  inches  long  is  your  step?  how  many  feet  long? 
how  many  yards  long? 

12.  How  many  inches  long  and  wide  is  your  schoolroom? 

13.  How  many  feet  long  is   your   schoolhouse?     how    many 
yards  long? 


MEASUREMENT  113 

14.  How    many    steps   wide    and    how   many   steps   long   is 
your  school    yard?    how  many    feet   wide   and    how   many   feet 
long?    how  many  yards? 

15.  How   many   steps   is  it  from  your  home  to  the  school- 
house?  how  many  feet?  how  many  yards?  how  many  rods  (1  rd.  = 
5*  yd.)? 

16.  How  far  can   you   walk   in   1   min.?      How  many   miles 
could  you  walk  in  1  hr.  at  the  same  rate? 

17.  How  far  is  it  from  your  home  to  a  neighboring  large  city? 
Answer  this  by  using  a  map  and  the  scale  given  with  it.     How 
long  would  it  take  you  to  walk  to  that  city  at  the  rate  of  walking 
in  problem  16? 

18.  How  long  will  it  take  a  train  to  run  from  your  nearest 
station  to  that  city  at  24  mi.  per  hour? 

19.  Using    the   scale    map  of    your    state    (see    Geography) 
find  the  length  and  the  width  of   your  state;   of  your  county; 
of  the  U.  S. 

20.  How  long  and  how  wide  is  a  two-cent  postage  stamp? 

21.  How  long  is  the  diameter  (distance  across)  of  a  copper 
cent   (see  Fig.  40)?  of  a  dime?    of  a  nickel?  of  a  quarter  dollar? 
of  a  half  dollar?  of  a  silver  dollar? 

22.  How  long  is  the  radius  of  each  of  these  coins? 

23.  Wrap  a  strip  of  paper,  or  a  string,  around  each  of  these 
coins  and  find  how  far  it  is  around  each. 

24.  What  kind  of  unit  do  you  need  to  measure  and  express 
long  distances?  medium  distances?   very  short  distances? 

NOTE.— Such  a  number  as  three-eighths,  or  f,  of  an  inch  means  three 
units  each  one-eighth  of  an  inch  long.  Such  a  unit  is  called  a  fractional 
unit,  and  such  numbers  as  |,  |,  £,  J|,  }•$,  ff,  are  all  said  to  be  numbers 
expressed  in  fractional  units.  Name  the  unit  of  each  of  the  six  frac- 
tions just  given. 

25.  How    long    is    the    distance    around    (circumference)    a 
bicycle  wheel  (wrap  a  string  around  the  wheel)?     How  long  is 
its  diameter? 

26.  Answer  the  same  questions  for  a  carriage  wheel;  for  the 
bottom  of  a  bottle ;  of  a  can ;  for  the  head  of  a  barrel ;  for  any 
other  circles  you  can  find. 


114  RATIONAL    GRAMMAR   SCHOOL    ARITHMETIC 

27.  Arrange  these  measures  thus: 


OBJECT 

DIAMETER 

CIRCUMFERENCE 

Silver  dollar 

Half  dollar 

4 

Quarter  dollar  . 

Nickel  

Dime 

Cent 

Bicycle  wheel 

Carriage  wheel 

Bottle 

Can 

Barrel  head 

Fill  out  columns  2  and  3  of  a  table  like  this  with  your  measurements 
and  keep  them  for  later  use. 


o 

B 

7 

* 

5 

4 

I 

.3 

2 

• 

• 

• 

'.:•: 

' 

1 

i    a   3    A  .5    e   7   a   9  10  it  ie 

;78.  Measuring  Surfaces. 

A  square  unit  is  a  square  I  unit 
long  and  1  unit  wide. 

1.  The  side  wall  of  a  room  is  9  ft. 
by  12  ft.     What  will  it  cost  to  lath  and 
plaster  the  wall  at  3</;  per  square  yard? 

2.  Using  your  own  measures  of  the 
side  and  the  end  walls  of  your  school- 
room, answer   the   same  question  for 
its  walls  and  ceiling. 

3.  How  many  square  feet  of  black- 
board surface  are  there  in  your  room? 

4.  How  many  square  inches  are  there  in  a  pane  of  glass  in  one 
of  your  windows?     How  many  square   feet   of   window   surface 
admit  light  into  your  room? 

5.  This  should  be  at  least  as  great  as  \  of  the  floor  surface  of 
your  room.     Is  it? 

6.  TV  of  an  inch  in  the  drawing,  Fig.  56,  stands  for  one  foot 
in  the   room.     Measure  and  find  the  number  of  square  feet  of 
plastered  surface  in  the  whole  interior  of  the   room,  deducting 
the  part  covered  by  the  baseboard  and  the  floor. 


Scale    ft"-!' 

FIGURE  55 


MEASUREMENT 


115 


Such  a  representation  of  the  room  as  that  of  Fig.  56  show- 
ing the  walls  and  the  ceiling  spread  out  on  a  flat  surface  is 
called  a  development  of  the  room. 


Ceiling 


End  Wall 


Side 


End  Wall 


Base  Board 


Floor 


7.  Draw  to  scale, 
from  your  own  meas- 
ures, the  development 
of   your  own   school- 
room. 

8.  In   Fig.   57   all 
the  different  kinds  of 
four-sided    plane  fig- 
ures are  represented. 
All    but    No.    7    are 
quadrilaterals.     What 
is  a  quadrilateral? 

9.  H  o  w     m  any 
pairs  of  parallel  sides 
have    1,    2,    3,    4,   5, 
and  6? 

10.  How    many 
pairs   of    equal    sides 
have  1,2, 3,  4, 5,  and  6? 

11.  How  many  square  corners  have  1,  2,  and  3? 

12.  Nos.    1,    2,    3,    and   4   are  parallelograms.      What   is    a 
parallelogram? 

13.  Nos.  1  and  2  are  rectangles.     What  is  a  rectangle? 


oase  Doara. 


Side  Wall 


FIGURE  56 


FIGURE  67 

14.  In  what  way  are  a  square  and  a  rhombus  alike?  unlike? 

15.  In  what  way  are  a  square  and  a  rectangle  alike?  unlike? 

Fig.  58  represents  a  plot  of  the  streets  and  blocks  of  a  part  of 
a  certain  city.  The  streets  are  all  75  ft.  wide  between  sidewalks. 
The  numbers  written  on  the  lines  indicate  their  lengths  in  feet. 


116 


RATIONAL    GRAMMAR   SCHOOL    ARITHMETIC 


The  dotted  lines  show  how  to  subdivide  the  areas  into  parts  for 
computation 


ji i 


FlGUKE   58 


Before  finding  the   areas   of  the  parts,  the  study  of  a   few 
forms  is  necessary.  , 

16.  What  is  the  area  of  the  rectangle  of  Fig.  58a? 


FIGURE  58a 

17.  Examine  the  parallelograms  and   the  rectangles  beneath 
them  and  find  the  areas  of  the  parallelograms. 

18.  If  the  length  of  any  parallelogram  and  its  distance  square 
across  (altitude)  are  given,  how  can  we  find  a  rectangle  with  the 
same  area  as  the  parallelogram? 

19.  Point  out  in  Fig.  58  a  rectangle  that  has  the  same  area 
as  the  parallelogram  $. 


MEASUREMENT 


117 


20.  The  length  of  a  parallelogram  is  b  ft.  and  its  altitude 
is  a  ft. ;  what  is  its  area? 

21.  Call  P  the  area,   ~b  the  length,  and  a  the  altitude  of  a 
parallelogram,  write  an  equation  to  show  how  you  would  use  b 
and  a  to  find  P. 

22.  If  $240  per  foot  of  frontage  on  both  3d  and  4th  streets 
was  paid  for  block  B  (Fig  58),  how  much  did  the  block  cost 
per  square  foot?  per  square  yard? 

23.  Find  the  area  in  square  yards  of   other  parallelograms, 
such  as  0,  P,  etc.,  of  Fig.  58. 


24.  Study  the  triangles  of  Fig.  59  and  find  to  what  parallelo- 
gram the  area  of  a  triangle  bears  a  simple  relation.     What  part  of 
the  area  of  the  parallelogram  equals  the  area  of  the  triangle? 

25.  Find  the  areas  of  the  triangles  A,  B,  C,  and  /),  Fig.  59. 

26.  Draw  a  scalene  triangle  and  show  how  to  complete  a  par- 
allelogram on  it  in  tlir^e  different  ways.     What  part  of  the  par- 
allelogram equals  the  triangle  in  each  case? 

27.  How  can  you  find  the  area  of  any  triangle  when  you  know 
its  length  and  its  altitude  (=  shortest  distance  to  the  base  from 
the  opposite  corner)? 

28.  Calling  T  the  area  of  a  tri- 
angle, ABC,  b  its  base,  AB,  and  a 
its  altitude,  CE  (Fig.  60),  write  an 
equation  to  find  ^from  a  and  b. 

29.  Find  the  area  in  square  feet 
of  the  following  triangles  of  Fig.  58 


G 


I ;  J\  K\  L ;  abc. 
square   across 


The   altitude   of    a   trapezoid  is   the   distance 
between  the  two  parallel  sides  (called  the  bases). 

30.  Study  the  trapezoids  of  Fig.  61,  and  find  how  to  get  the 
length  of  EF,  the  line  connecting  the  middle  points  of  the  two 
non-parallel  sides,  from  the  lengths  of  -  the  two  bases.  After  c6m- 


118 


RATIONAL    GRAMMAR    SCHOOL    ARITHMETIC 


puting  EF,  find  for  each  trapezoid  a  rectangle  whose  area  equals 
the  area  of  the  trapezoid.  Find  the  areas  of  the  trapezoids  of 
Fig.  61. 


FIGURE  61 

DEFINITION.— The  lengths  a,  b,  and  c  of  the  last  trapezoid  are  called 
its  dimensions. 

NOTE. — |  the  sum  of  x  and  y  is  written  |  (x-\-y\  n  times  the  half 
sum  is  written  $n(x  -f-  y). 

31.  How  do  the  altitude  and  the  sum  of  the  bases  of  a  trapezoid 
compare  with  the  altitude  and  base  of  a  parallelogram  having  an 

v\"7 \        V "SA         area  equal  to  the  area  of  the 

E/\         X     \      '\      trapezoid  (Fig.  62)? 
FIGURE  62  32.  Supposing  a,  #,  and  c 

are  the  dimensions,  in  feet,  of  the  trapezoids  of  Fig.  62,  what 
are  the  areas  of  these  trapezoids? 

33.  Find  the  areas  in  square  feet  of  C\  of  D\  of  M\  of  N\ 
of  H  (Fig.  58). 

34.  Calling  Z  the  area  of  any  trapezoid,  whose  bases  are  b  and 
c  and  whose  altitude  is  «,  write  an  equation  showing  how  to  find  Z 
from  «,  5,  and  c. 

35.  Find  the  area  in  square  yards  of  A,  B,  F,  N,  0,  P,  S,  T, 
Fig.  58. 

§79.  Measuring  Volume  (Bulk)  and  Capacity. 

1.  What  is  the  capacity  of  a  square  cornered  box 
3"  x  4"  x  6"  (see  Fig.  63)?   3'  x  ±'  x  6'?  of  a  room  3  yd. 
x  4  yd.  x  6  yd.?     a  yd.  x  I  yd.  x  c  yards? 

2.  A  box-car  8'  x  34'  can   be  filled  with  wheat  to 
a  height  of  5  ft.     When  full  how  many  cubic  feet  of 
grain  does  it  hold? 

3.  If   a  bushel    of    wheat  =  f  cu.  ft.,  how  many 
bushels  does  the  car  hold? 

4.  Noticing  that  a  cubic  foot  of  grain  (not  ear-corn)  is  |  bu., 
make  a  rule  for  finding  the  number  of  bushels  a  wagoia-box  or 
a  granary  will  hold  when  full. 


FIGURE  63 


MEASUREMENT 


119 


5.  A  wagon-box  is  2'  x  4'  x  10'.  How  many  bushels  of  ear-corn 
will  it  hold  if  J-  cu.  ft.  =  1  bu.  of  ear-corn? 

0.  How  many  rectangular  solids  4"  x  3"  x  7",  will  fill  a  box 
16"  x  12"  x  28  inches?  (See  Fig.  04.) 

7.  A  box  30"  x  24"  x  12"  will  contain 
how  many  blocks  6"x  4"  x  3"  inches? 

8.  128    cu.  ft.  =•  1  cd.       How   many 
cords  in  a  straight  pile  of    wood  80  ft. 
x  4  ft.  x  4  feet? 

9.  The  volume  of  a  rectangular  bin  is  1500  cu.  ft.     If 


••**• 

FIGURE  64 


it  is 


25  ft.  long  and  6  ft.  deep,  what  is  its  width? 

10.  How  many  bricks  8"  x  4  "  x  2"  are  there 
in  a  regular  pile  218"  x  24"  x  48"  inches? 

11.  Measure   the  length,  the  width,  and  the 
height  of   your  schoolroom,  and  find  how  many 
cubic  feet  of  air  it  contains. 

12.  How  many  pounds  does  the  steel  T-beam 
of   Fig.   65  weigh   if    1   cu.   in.  of   steel  weighs 
4J  ounces? 

13.  A  steel  I-beam  24'  long  has  a  cross  section  of  thejiorm 
and  size  shown  in  Fig.  66.     How  much  does  it  weigh 

if  steel  weighs  486  Ib.  per  cubic  foot  (metal  1"  thick)? 

14.  A  water   tank   is   3'  x  0'  x  10'.       How    many 
cubic  feet  of  water  does  it  hold  when  full? 

15.  There  are  231  cu.  in.  in  a  gallon;  how  many 
gallons  does  the  tank  of  problem  14  hold? 

16.  How  many  cubic  inches  of  grain  does   a   box  hold  if  it 
is  8  in.  square  and  4|  in.  deep?  how  many  gallons? 

17.  How  many  cubic  inches  does  a  box  hold  if  it  is  16  in. 
square   and   S|-   in.   deep?    how  many  bushels  (2150.2  cu.  in.  = 
1  bushel)? 

18.  How    many    cubic    inches    does    a    box    hold   if    it     is 
9"  x  10"  x  12"?  about  what  part  of  a  bushel? 

19.  Answer   similar   questions   for   a  box    10   in.  square  and 
10J  in.  deep;  for  a  box  10"  x  12"  x  18". 


120 


RATIONAL    GRAMMAR   SCHOOL    ARITHMETIC 


.---->    To 

pView      «--« 

0             0 
Side  View 
o          o 

Sieve 

Scale  /4 
FIGURE  67 

§80.  Measuring  Weight. 

1.  Fig.  67  shows  the  top  and  side  views  of  a  sand  sieve  drawn 
to  a  scale    of    J.       Measure  the  drawing  and  with   the  aid   of 

the  scale  find  the  length, 
breadth,  and  depth  of  the 
sieve.  Answer  for  both  out- 
side and  inside  measures. 

2.  How  thick  is  the  stuff 
used  for  the  frame  of  the 
sieve? 

Two  sieves  were  made 
in  the  manual  training  shop 
according  to  the  drawings 
of  Fig.  67.  The  bottom  of 

one  was  covered  with  wire  netting   of  TV  in.  mesh,  and  of  the 

other  with  wire  of  -fa  in.  mesh. 

3.  A  6-in.  cube  of  soil  weighed  162  oz.     What  would  a  3-in. 
cube  of   the  same  soil  weigh?   a  2-in.  cube?   a  4-in.  cube? 

4.  A  4-in.  cube  of  natural  soil,  weighing  48  oz.,  was  thoroughly 
dried  and  then  found  to  weigh  38  oz.     What  is  the  weight  of  water 
in  the  natural  soil?     How  many  ounces  of  water  would  there  have 
been  in  a  cubic  foot  of  the  same  soil? 

5.  24  oz.  of   dry   sandy  loam   were   broken   up  and  rubbed 
through  a  coarse  sieve.     8   oz.  of  coarse  gravel  remained  on  the 
sieve.     What  part  of  the  loam  was  coarse  gravel? 

6.  The  16  oz.  that  passed  the  coarse  sieve  were  rubbed  through 
the  fine  sieve.     The  coarse  sand  that  remained  on  the  fine  sieve 
weighed  7-J-  oz.    What  part  of  the  24  oz.  of  loam  was  coarse  sand? 

7.  A  4-in.  cube  of  dry  soil  weighed  18  oz.     It  was  then  thor- 
oughly saturated  with  water  and  found  to  weigh  32  oz.     How 
much  water  would  a  cubic  foot  of  this  dry  soil  be  capable  of  hold- 
ing when  saturated? 

8.  Find  the  difference  between  your  weight  and  that  given  in 
the  table,  p.  18,  for  a  boy  or  girl  of  your  age. 

9.  Estimate  the  weight  of  your  book,  or  of  a  brick,  or  other 
object,  then  weigh  it  and  find  the  difference  between  the  true 
weight  and  your  estimate. 


MEASUREMENT  121 

10.  Find  the  cost  of  3f  Ib.  of  wire  nails  @  4  cents. 

11.  What  is  the  cost  of   1  Ib.  of  sugar  selling  22  Ib.  for  a 
dollar? 

12.  Postage  on  first-class  mail  matter  is  2^  an  ounce.     What 
would  the  postage  be  on  a  2f  Ib.  package  of  first-class  matter 
(16  oz.  =  1  pound)? 

13.  Find  the  cost  of  2f  Ib.  butter  @  32  cents. 

14.  New  York  merchants  bought  in  1  da.  the  following : 

6324  Ib.  butter  at  an  average  price  of  22£  cents ; 
1988  Ib.  cheese  at  an  average  price  of  14f  cents; 
6840  Ib.  sugar  at  an  average  price  of  4f  cents; 
2780  Ib.  sugar  at  an  average  price  of  4.8  cents. 

Find  the  total  weight  and  the  total  cost. 

15.  Find  the  total  number  of  pounds  and  the  total*  value  of 
these  purchases : 

650  Ib.  cut  loaf  sugar  @  $5.74  per  hundredweight  (cwt.); 
825  Ib.  granulated  sugar  @  $4.80  per  hundredweight; 
400  Ib.  powdered  sugar  @  $5.24  per  hundredweight; 
700  Ib.  confectioners   sugar  @  $4.89  per  hundredweight; 
350  Ib.  extra  white  sugar  @  $4.78  per  hundredweight. 

16.  Find  the  weight  and  total  value  of  this  shipment: 

5400  Ib.  plain  beeves  @  $5.70  per  hundredweight; 

6375  Ib.  choice  beeves  @  $5.80  per  hundredweight; 

8450  Ib.  fair  beeves  @  $4.90  per  hundredweight; 

8625  Ib.  medium  beeves  @  $4.60  per  hundredweight; 
40,700  Ib.  veal  calves  @  $6.50  per  hundredweight; 
43,000  Ib.  western  steers  @  $8.50  per  hundredweight; 

8450  Ib.  Texas  steers  @  $3.70  per  hundredweight; 
25,200  Ib.  beef  cows  @  $2.85  per  hundredweight. 

17.  How  many  tons  did  the  purchases  of  problem  14  weigh 
(2000  Ib.  =  1  T.)?    How  many  tons  in  the  shipment  of  problem  16? 

18.  A  troy  ounce  of  pure  gold  is  worth  $20.67.     How  much 
is  a  troy  pound  of  pure  gold  worth  (12  troy  oz.  =  1  troy  pound)? 


122  RATIONAL    GRAMMAR    SCHOOL   ARITHMETIC 

19.  An   avoirdupois   pound    equals    l-f^    of    a   troy   pound. 
About  what  is  the  value  of  an  avoirdupois  pound  of  gold? 

NOTE.— Add  to  the  value  of  a  troy  pound  of  gold  31  times  ,  *4  of  its 
value. 

20.  An  avoirdupois  pound  =  10  avoirdupois  oz. ;  what  is  the 
value  of  an  avoirdupois  ounce  of  gold? 

21.  What  is  the  value  of  your  weight  in  gold?     (Your  weight 
is  given  in  avoirdupois  pounds.) 

22.  A  grain  dealer  received  during  January  130  carloads  of 
grain,   averaging  25^  T.  each.      Counting  GO  Ib.  to  the  bushel, 
how  many  bushels  did  he  receive  during  the  month? 


§81.  Measuring  Temperature. 

On  the  Fahrenheit  thermometer  the  point  where  water  begins 
to  freeze  is  marked  32°,  and  the  point  where  water  begins  to  boil 
is  marked  212°.  (See  Fig.  12,  p.  23.) 

1.  How  many  degrees  are  there  between  the  boiling  point  and 
the  freezing  point? 

2.  At  11  p.m.  on  a  certain  date  a  thermometer   read    32°. 
The  mercury   then  fell   £°  an  hour  for   4   hr.      What  was  the 
reading  at  3  a.m.  the  next  day? 

3.  The    mercury  then  fell  1°  an  hour  for  5  hr.     What  was 
the  reading  at  8  a.m.? 

4.  At  3  a.m.  on  a  certain  day  the  reading  was  26°.      The 
mercury  fell   2°  an  hour  for  5  hr.      What  was  the  reading  at 
8  a.m.? 

5.  On  a  certain  date  the  reading  was  10°,  and  the  mercury 
fell  on  the  average  lf°  an  hour  for  7  hr.     What  was  the  reading 
at  the  end  of  the  7  hours? 

6.  It  then  fell  1^°  an  hour  for  8  hr.     What  was  the  reading 
then? 

7.  The  mercury  fell  from  the  reading  12°   above  zero  to  3° 
below  zero.     How  many  degrees  did  it  fall? 

8  We  might  write  readings  above  zero  thus :  A.  12°;  A.  6°; 
A  32°;  and  readings  Mow  zero  thus:  B.  3°;  B.  6°;  B.  30°. 


MEASUREMENT  123 

How  many  degrees  does  the  mercury  fall  from  the  first  of  these 
readings  to  the  second? 

(1)  A.    8°  to  A.  3°;   (4)  A.  5°    to  B.    7°;   (7)  B.    2°  to  B.  11°; 

(2)  A.    8°toB.  3°;    (5)  A.  2°    to  B.  12°;    (8)  B.  7£°  to  B.  13°; 

(3)  A.  30°  to  B.  2°  ;    (6)  A.  4|°  to  B.  4|°  ;    (9)  B.  9^°  to  B.  12|°. 

9.  How  many  degrees  of  rise  or  fall  are  there  from  the  first  of 
these  readings  to  the  second?  If  the  change  is  a  rise,  mark  it  R; 
if  a,  fall,  mark  it  F.  : 

(1)  B.  7°    to  B.  2°;  (5)  A.  9°    to  B.     9°;  (9)  A.  6|°to  A.    12°; 

(2)  B.  2i°  to  B.  1°;  (0)  A.  2|°  to  B.  2|°  ;  (10)  A.  3°  to  B.  30*°; 

(3)  B.  2i°  to  A.  1°;  (7)  B.  15°  to  B.  6i°  ;  (11)  A.  18°  to  A.  67|°  ; 

(4)  A.  3£°  to  B.  1°  ;  (8)  B.  15°  to  A.  6^°  ;  (12)  B.  22£°  to  A. 


10.  Instead  of  writing  an  A.  for  ''above  zero,"  readings,  it  is 
customary  to  use  the  sign  (+).  What  sign  would  you  then  sug- 
gest for  "below  zero"  readings?  Tell  whether  a  change  from  the 
first  to  the  second  of  these  readings  denotes  a  rise  or  a  fall  in  each 
case  and  by  how  much? 

(1)  +  16°  to  +  100°  ;   (4)   +  32°  to  -    3°  ;   (7)   -  18°  to  -  34°  ; 

(2)  +32°  to  4-212°;   (5)   -I-  16°  to  -  17°  ;   (8)  -18°to-6|°; 

(3)  +    2°  to  +  161°;   (6)  +    8°  to  -  30°;  (9)  -     6°  to  -  2 


11.  The  12  o'clock  (noon)  readings  for  4  successive  days  were 
as  follows  :  +  82|°  ;  +  78f  °  ;  +  61^°  ;  +  534°.    What  is  the  average 
of  these  12  o'clock  readings. 

12.  Find  the  average  of  these  9  a.m.  readings  for  G  da.  :  +  9°  ; 
4-4°;  +5°;  +  12$°;+  14±°  ;  +  9J°. 

13.  Find  the  average  of  these  6  readings:  -  4°;  -  6°;  -  2°; 
-5°;  -  13°;  -  12°. 

14.  Find  the  average  of  these  2  readings:    -r  8°  and  —2°. 
Show  on  Fig.  12,  p.  23,  what  point  on  the  thermometer  is  midway 
between  the  readings  +8°  and  -  2°. 

15.  What  point  is  midway  between  the  readings  +13°  and 

-  5°?   between  the  readings  +  4°  and  -  6°?    +  12°  and  -  12°? 

-  4°  and  -  12°? 


124  RATIONAL    GRAMMAR    SCHOOL   ARITHMETIC 

§82.  Measuring  Time. 

The  primary  unit  used  in  measuring  time  is  the  mean  solar 
day.  This  day  is  the  average  time  interval  during  which  the 
rotation  of  the  earth  carries  the  meridian  of  a  place  eastward  from 
the  sun  back  around  to  the  sun  again.  It  is  the  average  length 
of  the  interval  from  noon  to  the  next  noon. 

1.  If  the  time  piece  is  running  correctly,  how 
many  times  does  the  short  hand  (the  hour  hand) 
of  a  watch  or   clock    turn    completely   around 
from  noon  to  the  next  noon? 

2.  How  much  time  is  measured  by  one  com- 
plete turn  of  the  hour  hand? 

3.  What  part  of  a  day  of  24  hr.  is  measured 
by  the  rotation  of   the   hour   hand    from   XII 
to  VI?  from  XII  to  III?  from  XII  to  I?  to  II? 

FIGURE  68  t()  IXJ? 

4.  What  name  is  given  to  the  time  interval  in  which  the  hour 
hand  moves  from  XII  to  I? 

5.  For  measuring  shorter  periods  the  motion  of  the  long  hand 
(the  minute  hand)  is  used.     What  part  of  a  day  is  measured  by 
the  movement  of  the  minute  hand   from    XII    around  to   XII 
again?  from    XII  to  VI?   from    XII  to  III?   from  XII  to    IX? 
from  XII  to  I? 

6.  What  name  is  given  to  the  interval  of  time  required  for  the 
long  hand  to  make  one  complete  turn? 

7.  How  many  times  does  the  minute  hand  turn  around  while 
the  hour  hand  turns  around  once? 

8.  The  small  and  rapidly  moving  hand  covering  the  VI  of  the 
watch  face  is  the  second  hand.     How  much  time  is  measured  by 
one  whole  turn  of  the  second  hand? 

9.  Over  how  much  space  does  the  tip  of  the   minute  hand 
move  while  the  second  hand  moves  around  once? 

10.  The  second  hand  circle  is  divided  into  how  many  equal 
parts?     What  name  is  given  to  the  time  interval  in  which  the 
second  hand  moves  over  one  of  these  equal  spaces? 

11.  How  many  times  does  the  minute  hand  turn  around  in  one 


MEASUREMENT 


125 


day?  in  a  week?   in  a  month  of  30  da.?    in  365  da.   (1  common 
year)  ? 

12.  Answer  the  same  questions  for  the  second  hand. 

13.  There  are  about  365^  da.  in   1  yr.    If  the  hour  and  the 
minute  hands  of  a  watch  start  at  XII  and  the  second  hand  at  60 
at  the  beginning  of  a  year,  how  will  the  hands  point  at  the  instant 
the  year  ends,  if  the  watch  runs  correctly  and  without  stopping? 

14.  Answer  question  13  supposing  the  length  of  the  year  to 
be  365  da.  5  hr.  48  min.  46  seconds. 


§83.  Measuring  Land. 

1.  A  section  of  land  is  a  tract  320 
rd.,  or  1  mi.,  square.  It  is  usually 
divided  into  halves,  quarters,  eighths 
and  sixteenths,  as  shown.  How  many 
square  rods  make  a  section  of  land? 


w 


2.  An    acre 
many   acres    in 


160    sq.    rd. 
of 


how 
land? 


J  Brown 

I 

T) 

CO 

| 

§ 

WH  Dugan 

"* 

-\ 

? 

A  .S.Park 

1 

\ 

J.M5mith 

SECTION  OF  LAND— FIGURE  69 


a    section 
in  a  quarter  section? 

3.  A  section  of  land  is  divided  up  into  farms,  as  shown  in  the 
cut;  how  many  acres  are  there  in  the  farm  belonging  to  A.  S.  Park? 
to  H.  S.  Barnes?  to  J.  Brown?  H.  A.  Dryer?  J.  M.  Smith?  P.  S. 
Mosier?  J.  S.  Doe? 

4.  Point  out  the  south  half  of  the  section;  the  east  half;  the 
1ST  i;  the  W  4. 

5.  Point  out  the  SE  £;  the  NW  i;  the  SW  i;  the  XE  ±. 

6.  Point  out  the  N  £  of  the  SE  i;  the  E  -J-  of  the  SE  ±;  the 
W  J  of  the  NE  i;  the  S  £  NW  i;  the  W  |  EJ. 

7.  Point  out  the  SW  i  of  the  SE  i;  the  NE  i  SW  £;  the 
NW  i  NW  i;  the  NE  i  SW  ±. 

8.  The  farm  of  E.  Miner  would  be  described  as  the  N  -J  E  -J- 
E  -J-  NE  i ;  describe  the  farms  of  the  8  other  owners  of  this  section. 

9.  How  many  rods  of  fence  would  be  required  to  enclose  the 
section  and  separate  it  into  farms  as  shown? 

10.  How  many  rods  long  is  a  ditch  starting  from  the  NE  corner 
of  H.  A.  Dryer's  farm,  thence  running  due  south  to  H.  S.  Barnes's 


126 


RATIONAL    GRAMMAR    SCHOOL    ARITHMETIC 


N  line,  thence  duo  W   to  Barnes's  W  line,  thence  S  to  J.  M. 
Smith's  N  line,  thence  due  W  across  Park's  farm  to  his  W  line? 
11.  Each  small  square  in  Fig.  70  represents  a  section.     The 

large  square  represents  a  township;    how  long 

is  a  township?  how  wide? 

12.  How  many  square  miles  in  1  Tp.?  how 
many  acres? 

13.  Point  out  in  Fig.  70  the  following: 

(1)  NW  i  Sec.  33. 

(2)  S  i-  Sec.  17 ;  NE  i  Sec.  17 ;  E  ^  NE  -j 

TOWNSHIP— FIGURE  70     gec    -^ 

(3)  NW   i  Sec.   21 ;    NE  ±  NW  }-  Sec.  21 ;    SE  ±  NW  i 
Sec.  21. 

14.  How  many  farms  of  100  A.   each  could    be  made    of   a 
township? 

15.  How  long  and  how  wide  is  a  square  160-acre  farm? 


§84.   Plotting  Observations  and  Measurements. 

1.  The  hourly  temperatures  from   G   a.m.  to  0  p.m.  Decem- 
ber 2G,  1902,  were: 

6  a.m.       7  a.m.       8  a.m.      9  a.m.      10  a.m.      11  a.m. 

7°  7°  10°  10°  12°  13° 

12m.     1p.m.     2p.m.     3p.m.     4p.m.     5p.m.     G  p.m. 
14°       14°          15°          16°  15°          14°          14°' 

Draw  13  equally  spaced  vertical  parallels 
one  for  each  hour.  Draw  a  horizontal  line 
as  OX  across  the  parallels.  Using  £"  to  repre- 
sent 1  ° ;  measure  off  on  the  first  vertical  the 
distance  01  to  represent  7°;  on  the  second 
vertical  the  distance  7-2  to  represent  the 
second  7°;  on  ihe  third  parallel  measure  off 
the  distance  10°,  and  so  on.  Mark  the  top  of 
each  measured  vertical  distance  with  a  dot. 
FIGURE  71  Draw  lines  connecting  these  dots  as  in  Fig.  71. 

This  is  called  plotting  the  readings. 
This  line  shows  the  temperature  change  during  the  12  hrs. 


MEASUREMENT  127 

This  plotting  is  very  much  aided  by  the  use  of  cross-lined 
paper,  ruled  into  small  squares.  A  horizontal  side  of  one  of  the 
small  squares  might  be  used  to  represent  1  hr.  and  a  vertical  side 
to  represent  1°  of  temperature. 

NOTE. — Pupils  in  arithmetic  should  have  notebooks  containing 
several  pages  of  squared  paper  for  such  work  as  this. 

2.  Eead  the  out-door  temperatures  from  hour  to  hour  at  your 
schoolhouse  and  plot  your  readings  as  above. 

3.  The  average  hourly  temperatures  from  6  a.m.   to  6  p.m., 
December  26  to  January  2,  were: 

6  a.m.       7  a.m.       8  a.m.       9  a.m.      10  a.m.       11  a.m. 
21.5°         22.1°        22.2°         22.2°        22.5°  23.2° 

12  m.    1  p.m.    2  p.m.     3  p.m.    4  p.m.     5pm.     G  p.m. 
23.9°        24°        24.5°       25.2°         25°       24.7°       24.3° 

Plot  these  readings  on  squared  paper,  or  by  measurement,  as 
above  in  figure  71.  Plot  the  tenths  by  estimate. 

4.  Does  the  line  for  the  averages  for  a  week  agree  in  a  general 
way  with  the  line  for  December  26?     What  does  a  comparison  of 
the  two  lines  show? 

This  table  gives  the  heights  in  feet  and  weights  in  pounds  of 
boys  and  girls  of  ages  given  in  the  first  column : 

HEIGHT  IN  FEET  WEIGHT  IN  POUNDS 

AGE  BOYS  GIRLS  BOYS  GIRLS 

0  1.6 

2  2.6 

4  3.0 

6  3.4 

9  4.0 

11  4.4 

13  4.7 

15  5.1 

17  5.4 

18  5.4 
20  5.5 

5.  Using  squared  paper,  or  making  a  drawing  such  as  is  indi- 
cated in  Fig.  72,  plot  the  numbers  in  the  first  column  horizontally 
and  those  in  the  second  column  vertically.      Draw  free-hand  a 


1.0 

7.1 

6.4 

2.6 

25.0 

23.5 

3.0 

31.4 

28.7 

3.4 

38.8 

35.3 

3.9 

50.0 

47.1 

4.3 

59.8 

56.6 

4.6 

75.8 

72.7 

4.9 

96.4 

89.0 

5.1 

116.6 

104.4 

5.1 

127.6 

112.6 

5.2 

132.5 

115.3 

128 


RATIONAL    GRAMMAR    SCHOOL    ARITHMETIC 


Ac 

es 

f 

JfU 

5 

X 

.346       9     II    13     5  171830 

FIGURE  72 

smooth  line  through  all  the  plotted  points  and  obtain  the  curve  for 
growth  in  stature  of  boys.  At  what  age  do  boys  cease  growing 
rapidly  in  height? 

6.  Using  the  numbers  of  columns  1  and  3  obtain  a  similar 

curve  for  girls.     At  what  age  do  girls  cease 

growing  rapidly  in  stature? 

7.  Compare   the   two    curves   and    note 
when  boys  and  when  girls  grow  fastest? 

8.  Using  the  numbers  of  columns  1  and 
4  draw   the  curve   for   growth   of   boys   in 
weight. 

9.  Use   numbers   of    columns    1    and   5 
similarly  for  girls. 

10.  Compare  the  curves  of  problems  8  and  9  and  note  any 
similarities   or   differences    in    the    two  curves.      What  do   the 
peculiarities  of  these  curves  show? 

11.  Twelve  different  rectangles  each  12  in.  long  have  the  widths 
given  in  the  first  line  and  the  areas  in  the  second  line. 

Widths  in  inches..!"  2"  3"  4"  5"  6"  7"  8"  9"  10"  11"  12" 
Areas  in  sq.  in.. ..12  24  36  48- CO  72  84  96  108  120  132  144 

Two  lines,  OX  and  OY,  are  drawn  at  right 
angles.  12  equally  spaced  lines  are  drawn  par- 
allel to  OY.  The  horizontal  distances  01,  02, 
03  and  so  on  denote  the  heights,  1",  2",  3'  and 
so  on.  On  a  scale  of  -J"  (or  the  vertical  side 
of  a  small  square)  to  12  sq.  in.  of  area,  mark 
off  the  lengths  la,  2b,  3c,  and  so  on,  to  denote 
the  numbers  in  the  second  line  above.  This 
gives  the  points  a,  b,  c,  and  so  on. 

Make  the  construction   to  the   scale  indi- 
cated or  to  some  other  convenient  scale,  and 
place  the  straight  edge  of  a  ruler  along  the  points,  such  as  a,  b, 
c,  and  so  on.     On  what  kind  of  line  do  the  points  seem  to  lie? 

12.  The  bases  of  12  triangles  are  each  16  in.  long  and  the  alti- 
tudes (heights)  and  the  areas  in  square  inches  are: 

V      2"      3"      4"      5"      6"      7"      8"      9"     10"     11"     12" 

8   16   24   32   40   48   56   64   72   80   88   96 


FIGURE  73 


MEASUREMENT 


129 


Using  any  convenient  scale,  plot  these  data.    On  what  kind  of 
line  do  all  the  points  lie? 

13.  The  lengths  of  the  sides  and  the  areas  in  square  inches  of 
12  squares  are : 

Sides.. I"    2"    3"    4"    5"    0"    7"    8"    9"    10"    11"    12" 
Areas..  1      4     9     1G    25    30    49    G4    81    100     121    144 

Using  any  convenient  horizontal  and  vertical  scales  plot  these 
observations.     Do  the  plotted  points  lie  on  a  straight  line? 

14.  Plot   these   data    and    draw  a    smooth    free-hand   curve 
through  the  plotted  points : 


DATE  OF 
CENSUS 

POPULATION  OF 
U.  S.  IN  MILLIONS 

DATE  OF 
CENSUS 

POPULATION  OF 
U.  S.  IN  MILLIONS 

1790 

3.9 

1850 

23.2 

1800 

5.3 

1860 

31.4 

1810 

7.2 

1870 

38.6 

1820 

a.6 

1880 

50.2 

1830 

12.9 

1890 

02.  6 

1840 

17.1 

1900 

76.3 

Plot  dates  on  horizontal,  and  populations  on  vertical,  lines. 

Draw  a  smooth  free-hand  curve  through  the  plotted  points. 
By  continuing  the  curve  can  you  predict  about  what  the  popula- 
tion will  be  in  1910? 

Is  the  line  through  the  plotted  points  a  straight  line? 


§85.  Measuring  By  Hundredths.     PERCENTAGE 


1.  What  is  T-J  o  of  500  mi.? 


of  500  mi.? 


2.  What  is 


of  200  A.?          of  200  A.? 


of  500  miles? 
of  200  acres? 


NOTE.  —  First  find  ^Q. 


3.  What  is  ^A  of  100  ft.?  T%  of  100  ft.?  ^j  of  100  feet? 

4.  -J  of  anything,  equals  how  many  hundredths  of  it? 

5.  How  many  hundredths  of  anything  are  the  following  frac- 
tional parts  of  it? 

i;  t;  i;  1;  !; 


~s  >  ro  5 


130  RATIONAL    GRAMMAR   SCHOOL    ARITHMETIC 

DEFINITION. — One  hundredth  is  often  written  \%  and  read  1  per  cent. 
The  sign  (% )  is  a  short  way  of  writing  "one  one-hundredth. "  Per  cent  is 
a  short  way  of  speaking  "one  one-hundredth." 

6.  Kead  and  give  the  meaning  of  the  following : 

2%;  6%;  8%;  12%;  12*%;  25%;  33*%;  87*%;  100%. 

7.  Give  the  simplest  fractional  equivalents  of  these  per  cents : 
50%;  25%;  75%;  12*%;  33*%;  66|% ;  20%;  40%;  80%. 

8.  How  many  Ib.  are  50%  of  8  lb.?  of  18  lb.?  of  24  pounds? 

9.  How  many  square  feet  are  25%  of  16  sq.  ft.?  of  48  sq.  ft.? 
of  88  sq.  ft.?  of  400  square  feet? 

10.  Heating  an  iron  rod  100  in.  long  increased  its  length  2%. 
How  many  inches  was  the  length  increased? 

11.  One  boy  threw  a  stone  100  ft.,  and  another  threw  it  12% 
farther.     How  many  feet  farther  did  the  second  boy  throw  the 
stone? 

12.  Referring  to  Fig.  22  (page  45)  point  out  the  following 
per  cents  of  it:  20%  ;  40%  ;  60%  ;  80%  ;  2%  ;  4%  ;  10%  ;  100%. 

13.  Draw  a  square  and  divide  it  by  a  line  so  as  to  show  50% 
of  it;  25%  of  it;  75%  of  it;  12*%  of  it. 

14.  Similarly,    show   the   same    per  cBnts    using  a  circle;  a 
rectangle. 

15.  Draw  a  line  of  any  length  and  show  33*%  of  it;  66f  %  of 
it;  100%  of  it;  12*%  of  it;  10%  of  it;  20%  of  it. 

16.  Draw  a  square  and  divide  it  into  small  rec- 
tangles as  shown  in  the  figure.  Point  out  the  following 
per  cents  of  it : 

FIGURE  74         25%;  33*%;  50%;  66f  % ;  75%;  12*%;  100%. 

17.  A  pair  of  shoes  costing  $2  was  sold  at  a  gain  of  50%. 
What  was  the  amount  gained? 

18.  A  boy  was  50  in.  in  height  2  years  ago.     Since  then  he 
has  grown  4%  higher.     How  many  inches  taller  is  he  now?     How 
tall  is  he  now? 

19.  The  height  of  a  14  year  old  boy  is  60  in.,  and  a  girl  of  the 
same  age  is  2%  taller.     How  tall  is  the  girl? 


MEASUREMENT  131 

20.  An  umbrella  was  marked  $1.50,  and  was  sold  at  a  reduc- 
tion of  20%.     At  what  price  did  it  sell? 

21.  Eighty  per  cent  of  the  cost  of  a  girl's  suit  is  $4.00.     What 
is  1%  of  the  cost?     What  is  the  whole  cost,  or  100%  of  the  cost? 

22.  An  article  sold  for  $4.00  after  having  been  marked  down 
20%.     What  was  the  price  before  it  had  been  marked  down? 

23.  A  lot  was  sold  for  $800,  which  was  £0%  more  than  it  cost 
the  man  who  sold  it.     How  much  did  it  cost  him? 

NOTE.— The  60%  means  60%  of  what  it  cost  the  seller.  This  cost  is 
how  many  per  cent  of  itself?  $800  is,  then,  how  many  per  cent  of  the 
original  cost.  1%  of  the  original  cost  equals  what?  100%  of  this  cost 
equals  what? 

24.  A  thermometer  reading  was  120°,  which  was  .20%  higher 
than  the  reading  taken  10  min.  before.     What  was  the  previous 
reading? 

NOTE.— In  such  questions  as  this  answer  first  the  question,  "20%  of 
what?"  In  this  case  20%  of  the  previous  reading. 

25.  A  man  sold  his  farm  for  $12,000,  which  was  25%  more 
than  he  paid  for  it.     What  did  he  pay  for  it? 

26.  This  year  a  flat  rented  for  $55  a  month,  which  is  10% 
more  than  it  rented  for  last  year.     For  what  did  it  rent  last  year? 

27.  Out  of  80  games  played  by  a  championship  team,  20  were 
lost.     What  per  cent  of  the  games  were  lost?     What  per  cent  were 
won? 

28.  The  chest  measure  of  a  boy  at  the  close  of  the  school  year 
was  27£  in.,  which  was  10%  greater  than  at  the  beginning  of  the 
year.     What  was  the  boy's  chest  measure  at  the  beginning  of  the 
year? 

29.  A  man  paid  $18  for  the  use  of  $300  for  a  year.     What  per 
cent  was  the  sum  he  paid  of  the  sum  he  borrowed? 

§86.  Simple  Interest. 

An  extensive  use  of  the  system  of  measurement  by  hnndredths 
is  made  with  problems  in  Interest. 

DEFINITION. — Interest  is  money  to  be  paid  for  the  use  of  money.  Two 
per  cent  interest  means  that  2/  is  to  be  paid  every  year  for  the  use  of  1000 
or  $1 ;  $2  for  the  use  of  $100,  and  so  on  at  this  rate.  What  then  would 
6%  interest  mean?  4%  interest?  10%  interest? 


132  RATIONAL    GRAMMAR   SCHOOL    ARITHMETIC 

The  amount  of  money  to  be  paid  each  year  for  the  whole  sum  bor- 
rowed is  called  the  Interest  for  1  year. 

1.  What  is  the  interest  on  $200  at  4%  for  1  year? 

2.  What  is  the  interest  at  6%  on  $1  for  1  year? 

NOTE. — In  computing  interest  a  year  means  12  months  of  30  days  each. 

3.  What  is  the  interest  at  6%  on  $1  for  ^  yr.,  or  6  mo.?  for 
2  mo.?  for  1  month? 

4.  What  is  the  interest  at  6%   on  $1  for  3  mo.?   for  5  mo.? 
for  7  mo.?  for  14  mo.?    for  16  mo.?   for  34  mo.?   for  any  number 
of  months? 

5.  From  the  answer  to  problem  2,  find  the  interest  at  6%  for 
1  yr.  on  $8;, on  $25;  on  $48;  on  $85;  on  $124;  on  $450. 

6.  If  you  know  the  interest  at  6%  for  1  yr.  on  $1,  how  can 
you  find  the  interest  at  6%  for  1  yr.  on  any  number  of  dollars? 

7.  From  the  last  answer  to  problem  5,  find  the  interest  at  6% 
on  $450  for  2  yr. ;    for  3  yr. ;   for  4£  yr. ;    for  12£  yr. ;   for  any 
number  of  years. 

8.  Using  the  third  answer  to  problem  4,  find  the  interest  at  6% 
for  7  mo.  on  $48;  on  $450;  on  $75. 

9.  Tell  how  to  find  the  interest  at  6%  on  any  number  of  dol- 
lars for  any  number  of  months. 

'  10.  The  interest  on  a  certain  sum  of  money  for  1  yr.  at  6%  is 
$24.  What  is  the  interest  on  the  same  sum  for  the  same  time  at 
12%?  at  18%?  at  30%?  at  60%?  at  1%?  at  3%?  at  2%? 

11.  f  of  the  interest  on  a  sum  of  money  for  a  given  time  at  6% 
equals  the  interest  on  the  same  sum  for  the  same  time  at  what 
rate  per  cent? 

NOTE. — |  of  a  number  may  be  easily  found  by  subtracting  from  it  |  of 
itself.     How  may  §  of  a  number  be  found  similarly? 

12.  Knowing  the  interest  on  any  sum  of  money  at  6%,  how 
can  you  quickly  find  the  interest  on  the  same  sum  for  the  same 
time  at  7%?  at  8%?  at  11%?  at  15%?  at  any  rate  per  cent? 

13.  Find  the  interest  on  $1200  at  6%  for  2£  yr. ;  for  2f  years. 

14.  A  man  has  $350  in  a  bank,  which  pays  3%  interest.     To 
how  much  interest  is  the  man  entitled  if  his  money  has  been  in 
the  bank  2  yr.  and  4  mo.?  6  yr.  and  9  mo.?  8  yr.  and  2  months? 


COMMON  USES  OF  NUMBERS 

§87.  Pressure  of  Air.  ORAL  WORK 

1.  When  a  glass  is  filled  level  with  water,  covered  with  a  piece 
of  writing  paper,  and  carefully  inverted,  why  does  not  the  water 
fall  out  when  the  glass  is  held  mouth  downward? 

2.  When  a  soft  leather  sucker  is  moistened  and  spread  out  on 
a  smooth,  flat  surface,  why  does  it  cling  to  the  surface  even  when 
the  sucker  is  raised  by  being  lifted  at  its  middle? 

3.  When  one  end  of  a  glass  tube  is  placed  in  water  and  you 
draw  with  the  mouth  at  the  other,  why  does  the  water  rise  in  the 
tube? 

4.  Fill  a  bottle  with  water,  leave  the  cork  out,  and  invert  the 
bottle  in  a  vessel  of  water.     Why  does  the  water  not  run  out? 

5.  When  a  glass  tube  with  one  end  closed  is  partly  filled  with 
mercury  and  the  open  end  dipped  under  the  surface  of  mercury 
in  a  cup,  why  does  not  all  the  mercury  run  out  of  the  tube? 

NOTE. — In  connection  with  5,  examine  a  mercurial  barometer. 

6.  When  a  sheet  of  thin  rubber,  or  paper,  is  held  over  the 
mouth  of   a   funnel,  and  the  air  is  sucked   out  of   the    funnel 
through  its  neck,  why  does  the  rubber  curve  inward?    Will  it  do 
this  in  ali  positions  of  the  funnel?     Why? 

These  experiments  show  that  the  air  presses  downward, 
upward,  and  in  all  directions  upon  surfaces.  Careful  measure- 
ments have  shown  that  the  pressure  of  the  air  on  every  square 
inch  of  surface  is  about  15  pounds. 

NOTE.  — This  pressure  is  equal  on  all  sides  of  surfaces,  upper  sides  and 
lower  sides,  outside  and  inside,  toward  the  right  and  toward  the  left. 

WRITTEN    WORK 

1.  Measure  the  length  and  the  width  of  the  cover  of  your  book 
and  find  the  downward  pressure  in  pounds  on  the  upper  surface  of 
your  book  when  it  lies  flat  upon  the  desk.  Why  is  the  book  so 
easily,  mo¥ed  about  .under,  t his  pressure?  -  .  ,  - 

133 


134  RATIONAL   GRAMMAR   SCHOOL   ARITHMETIC 

2.  Measure  and  compute  the  downward  pressure  of  the  air  on 
the  top  of  your  desk.      Can  you    lift  this    number  of  pounds? 
Why  can  you  lift  the  desk? 

3.  A  room  is  10'  x  20'  x  25'.     Find  the  pressure  of  the  air,  in 
pounds,  on  the  floor,  on  the  ceiling,  and  on  each  of  the  four  walls 
of  the  room.     Why  does  not  this  pressure  tear  the  walls  apart? 

4.  The  average  surface  of  the  human  body  equals  20.6  sq.  ft. 
What  is  the  total  pressure  of  the  air  on  the  outside   surface? 
Why  does  not  this  pressure  crush  the  body? 

5.  Measure  the  length  and  the  width  of  the  door  of  your  room, 
and  find  the  air  pressure  on  one  side  of  the  door.     Why  can  you 
open  and  close  the  door  so  easily? 

6.  Measure  and  find  the  air  pressure  in  pounds  on  the  outside 
surface  of  a  pane  of  window  glass.     Why  does  not  this  pressure 
break  the  glass? 

§88.  Passenger  and  Freight  Trains. 

1.  A  fast  train  runs  from  Chicago  to  a  station  356.4  mi.  dis- 
tant in  exactly  9  hr.     What  is  the  average  rate  (miles  per  hour) 
of  the  train? 

2.  The  train  left  Chicago  at  6:10  a.m.     At  what  time  did  it 
arrive  at  the  station? 

3.  Another  train  runs  10£  hr.  at  an  average  rate  of  36  mi.  per 
hour,  including  stops ;  how  far  does  it  run? 

4.  If  the  train  (problem  3)  started  at  2:  30  a.m.,  at  what  time 
did  it  reach  the  end  of  the  run? 

5.  A  traveler   went  by  rail  from   Chicago   to   Los   Angeles, 
California,  in  4  da.  19  hr.,  at  the  average  rate  of  37.4  mi.  an  hour, 
including  stops.     How  far  is  it  from  Chicago  to  Los  Angeles  by 
this  route? 

6.  A  train  ran  361  mi.  in  9J  hr.     What  was  its  average  rate? 

7.  A  freight  train  is  running  21  mi.  an  hour  while  a  brakeman 
on  top  of  the  cars  walks  toward  the  engine  at  the  rate  of  2-J  mi.  an 
hour.     How  fast  does  the  brakeman  actually  move  forward?  how 
fast,  if  he  walks  from  front  to  rear  at  the  rate  of  2-J  mi.  an  hour? 

8.  A  conductor  walks  from  the  front  to  the  rear  of  a  train  at 
the  rate  of  3£  mi.  an  hour  while  the  trait*  is  running  38  mi.  an  hour. 


COMMON"    USES    OF    NUMBERS  135 

At  what  rate  per  hour  does  the  conductor  actually  move  forward? 
At  what  rate  does  he  move  in  the  direction  of  the  running  train 
if  he  walks  from  the  rear  to  the  front  at  the  rate  of  3£  mi.  an 
hour? 

9.  How  may  the  conductor  (problem  8)  suddenly  change  his 
rate  of  motion  from  34£  mi.  to  4H  mi.  an  hour?     How  many 
miles  an  hour  does  this  change  of  rate  amount  to? 

10.  A  passenger  train  running  32  mi.  an  hour  meets  a  freight 
train  running  in  the  opposite  direction  on  a  parallel  track  18^  mi. 
an  hour.     At  what  rate  do  the  trains  approach  each  other? 

11.  At  what  rate  does  the  passenger  train  (problem  10)  pass 
the  freight  if  both  are  running  in  the  same  direction  on  parallel 
tracks? 

12.  A  boy  tries  to  overtake  a  street  car  by  running  up  behind 
it.    If  the  boy  runs  10  ft.  a  second  and  the  car  runs  6  ft.  a  second, 
how  soon  will  the  boy  overtake  the  car  if  he  is  now  20  ft.  behind 
it? 

13.  Two  friends,  coming  from  opposite  directions,  have  arranged 
to  meet  at  a  certain  railway  station.     Their  trains  are  running  at 
35  mi.  and  25  mi.  an  hour.    Not  allowing  for  time  lost  in  stops,  how 
soon  will  their  trains  be  together  at  the  station,  if  they  are  now 
30  mi.  apart  and  both  trains  reach  the  station  at  the  same  time? 

14.  A  passenger  train  is  made  up  of  a  postal  and  baggage  car. 
an  express  car,  3  common  coaches,  2  chair  cars,  a  dining  car,  and 
2  sleepers.    The  average  length  of  a  car  is  6 If,  and  the  length  of 
the  engine  and  tender  together  is  65'.     How  long  is  the  train? 

15.  The  weight  of  the  engine  (problem  14)  is  142,780  Ib. ; 
of  the  tender,  43,200  Ib. ;   and   the   average  weight  of  a  car  is 
83,480  Ib.     What  is  the  total  weight  of  the  train,  in  tons? 

16.  To  draw  a  train  on  straight,  level  track  at   a  speed  of 
40  mi.  an  hour,  requires  a  horizontal  pull  of  ?fo  of  the  weight 
of  the  train.     What  force  in  tons  would  be  needed  to  draw  the 
train  of  problem  15  at  a  speed  of  40  mi.  an  hour? 

17.  An  engine,  62'  long  and  weighing  240,000  Ib.,  draws  a 
tender,  22' long,  weighing  64,500  Ib.,  and  a  train  of  72  empty 
freight  cars,  averaging  36|'  in  length  and  32,700  Ib.  in  weight. 
How  long  is  the  train?  how  heavy? 


136  RATIONAL    GRAMMAR    SCHOOL    ARITHMETIC 

18.  If  it  requires  T/5Tr  of  the  weight  of  a  train  to  draw  it  on 
straight,  level  track  at  a  speed  of  20  mi.  an  hour,  how  many 
pounds  of  force  must  the  engine  of  problem  17  exert  to  draw  the 
train,  on  such  track,  20  mi.  an  hour?     How  many  tons  of  force? 

NOTE. — First  find  j^g  of  the  weight  of  the  train. 

19.  How  much  force  would  be  needed  to  draw  38  cars,  each 
weighing  16£  T.,  and  each  loaded  with  18£  T.  of  coal,  on  straight, 
level  track  at  a  speed  of  20  mi.  an  hour?     (See  problem  18.) 

20.  When  a  train  is  moving  5  mi.  an  hour  it  takes  a  horizontal 
force  of  about  T£¥  of  the  weight  of  the  train  to  draw  it  along 
straight,  level  track.     On  such  track  what  force  must  an  engine 
exert  to  draw  a  train  of  68  empty  freight  cars,  each  weighing 
34,300  lb.,  at  a  speed  of  5  mi.  an  hour? 

21.  Under  the  same  conditions  as  in  problem  20,  what  force 
would  be  needed  to  draw  a  train  of  38  cars,  each  weighing  34,000 
lb.  and  carrying  a  load  of  58,600  lb.  of  coal  in  addition  to  its  own 
weight? 

22.  A    railroad    company  purchased    6   locomotive    engines, 
weighing,  in  pounds,  142,780,  142,630,  158,670,  139,790,  146,890, 
and  138,960.     The  tenders  weighed,  in  pounds,  43,750,  44,200, 
45,280,  42,920,  43,650,  and  44,280.     The  cost  per  pound  of  the 
combined  weight  was  13f  <p.     What  was  the  total  cost? 

23.  The  greatest  horizontal  pulling  force  that  an  engine  can 
exert  on  dry,  unsanded  steel  track  in  starting  a  train  is  about 
•fs  of  the  part  of  the  engine  that  is  borne  by  the  driving  wheels. 
What  is  the  greatest  horizontal  pull  the  engine  of  problem  17  can 
exert  in  starting  a  train  on  such  track,  supposing  the  entire  weight 
of  the  engine  is  borne  by  the  driving  wheels? 

24.  Answer  a  similar  question  for  the  engine  of  problem  15, 
supposing  105,750  lb,  of  the  weight  of  the  engine  is  borne  by  the 
driving  wheels. 

§89.  Train  Despatched s  Report. 

Following  is  a  copy  of  a  train  despatch er's  sheet,  showing  the 
distance  between  stations,  the  schedule  time  and  the  exact  run- 
ning time  of  the  train,  the  number  of  the  cars  in  the  train,  the 
number  of  passengers  carried  xeach  trip  for  7  week  days. 


COMMON    USES   OF   NUMBERS  137 

PERFORMANCE  OF  TRAIN  No.  25,  FROM  JULY  IST  TO  STH,  1898. 


DlS- 

oTA*          T  *  wrrc>    oCHEDTJijE 

TION         M^         TIME 

1ST 

2D 

4TH 

5TH 

6TH 

7TH 

STH 

1  

0 

3:50 

3:50 

3:491 

3:50f 

3:50 

3:50^ 

3:50| 

3:51} 

2 

3.1 

3:55 

3:55 

3-55i 

3:56 

3:55 

3:55 

3:55^ 

3:56 

3 

5.5 

3:57 

3:57 

u  .wg 

3:58 

3  :58f 

3:57f 

3:581 

4  

7.9 

3:59 

3:59 

4:00 

4:00£ 

3:59* 

4:00| 

4:00 

4 

4:00 

5  .. 

12.0 

4:02 

4:02 

4:03f 

4  :03f 

4:03i 

4:04 

4:03 

4:03 

6  

17.0 

4:06 

4:06 

v"4 

4:073 

4 

4:07| 

•*-  •VV4 
4 

4:08J 

4:07£ 

7  

19.9 

4:09 

4:09 

4:10 

4  '09? 

4-081 

4:11 

4:10 

4:10 

8..    . 

24.5 

4:12 

4-1H 

4:14 

4:13 

4:12 

4:14$ 

4:14 

4:13| 

9  

27.6 

4:15 

2 

4:15 

4:14 

4:17a 

4:16 

4 

4:15 

10  

33.8 

4:20 

4:18 

4:211 

4:20 

4:18£ 

4:22 

4:201 

4:20| 

11 

38.7 

4:24 

4:22 

4 

4  :25 

4:24 

2 

4:22 

4:26i 

4:24| 

4:23 

12     

43.5 

4:28 

4:25| 

4  :27i 

4:26 

^V2 

4:30 

4:28 

4:27 

13 

50.5 

4:33 

4:34^ 

"•     J 

4-321 

4:31 

4-351 

4:33 

4:32 

14 

53.8 

4:38 

4-331- 

"        2 

4:35 

4:34 

«*"i 
4:38 

4:36 

4:354 

15  

55.5 

4:40 

4  :35£ 

4:40 

4:37| 

4:36 

4:40i 

4:372 

I 

4:37 

Number 

of  cars 

5 

7 

4 

6 

5 

^t        iV  ^ 

5 

-X.   »U  9    £ 

5 

5 

Passengers  carried  

201 

441 

11 

79 

107 

113 

118 

Running 

time 

Mi.  per  hour  (average)  .  . 

PROBLEMS 

1.  By  ''running  time"  is  meant  the  difference  between  the 
time  of  leaving*  station  1  and  the  time  of  arriving  at  station  15. 
Fill  out  the  blanks  in  the  line  "running  time." 

2.  Find  the  difference  between  "schedule  time"  and  the  time 
the  train  actually  reached  station  G  on  the  2d,  4th,  6th,  7th,  8th. 

3.  The  train  is  "on  time*'  if  it  reaches  a  station  just  at 
"schedule  time."     Find  how  much  the  train  was  ahead  of  or 
behind  time  in  reaching  station  12  on  each  of  the  7  da.     Mark 
"ahead  of  time"  results  with  an  "A,"  thus:  A.  2|  min.,  and 
"behind  time"  results  with  a  "B,"  thus:  B.  1£  minutes. 

4.  Make  and  answer  similar  questions  for  other  stations. 

5.  How  far  is  it  from  station  1  to  station  15?  from  2  to  9?  4  to 
11?  3  to  15?    Make  and  answer  similar  questions  for  other  stations. 

6.  What  is  the  difference  in  schedule  times  between  stations  1 
and  15?  2  and  9?  4  and  11?  3  and  15?  any  other  two  stations? 

*  Stops  are  so  short  that  leaving  and  arriving  times  are  regarded  as  the  same. 


138 


RATIONAL    GRAMMAR    SCHOOL   ARITHMETIC 


7.  From  your  answers  to  problems  5  and  6  find  the  average* 
rate  of  running  (in   miles  per   minute)  between   stations  1  and 
15,  by  a  train  running  exactly  on  schedule  time;  between  stations 
2  and  9;  4  and  11;  3  and  15. 

8.  Find  the  average  rates  of  running  between  the  same  stations 
(as  in  problem  7)  on  the  6th. 

9.  Find  the  average  rates  of  running  between  stations  1  and  15 
on  each  day  and  write  the  results  in  the  "mi.  per  hour"  line. 

10.  What  is  the  total  number  of  cars  drawn  during  the  whole 
period? 

11.  Find  the  total  number  of  passengers  carried. 

12.  Make  and  solve  other  problems  on  the  table. 

§90.  Areas  of  Common  Forms.     ORAL  WORK 

1.  If  one  side  of  an  inch  square  represents  80  rd.,  how  many 
square  rods  will  the  inch  square  represent?  how  many  acres? 

2.  In  a  drawing  of  a  rectangular  farm  to  a  scale  1  in.  =  80  rd., 
how  could  you  find  the  number  of  acres  in  the  farm? 

3.  If  a  square  field  containing  40  A.  is  cut  across  diagonally 
by  a  railroad,  how  many  acres  does  each  of  the  triangular  posts 
contain,  supposing  the  railroad  itself  covers  3  acres? 


FIGURE  75 

4.  Read  off  the  areas  of  the  cross-lined  portions  of  the  9  forms 
of  Fig.  75. 

*  "Average  rate"  here  means  distance  run  divided  by  time  of  running, 


COMMON    USES    OF    NUMBERS 


139 


PROBLEMS 

Each  inch  in  Fig.  76  represents  80  rods. 

1.  How  many 
acres  are  in  the 

tract   ABCDEt  \p 

What  is  the  farm 
worth  at  $95  per 
acre? 

2.  The  field  is 
entirely  enclosed, 
except  along  the 
river,  by  a  barb- 
wire  fence  having 
4  lines  of   wire. 
Barb  wire   weighs 
about  1  Ib.  per  rod. 
Find  the  cost  of  the 
wire   at   $3.50   per 
bale  of  100  pounds. 

3.  If  posts  cost 
35^  apiece  and  are 
set  2  rd.  apart,  find 
the     cost     of     the 

posts. 

w 

FIGURE  77 

4.  Answer  similar  questions  for  Fig.  77,  the  scale  being  the 
same  as  for  Fig.  76. 

5.  If  the  weight  of  a  certain  piece  of  inch  board  is  20  oz.,  and 
it  represents  the  area  of  a  20-A.  field,  how  many  acres  would  be 
represented  by  a  piece  of  the   same   board  weighing  37  ounces? 

6.  A  farmer  stepped  the  4  sides  and  the  diag- 
onal of  an  irregular  field  and  found  them  to  be 
60  rd.,  70  rd.,  75  rd.,  85  rd.,  and  80  rd.     He 
drew  on  a  board  a  4-sided  figure  (like  Fig.  78, 
but  larger)  to  represent  the  field.     The  sides  of 
the  drawing  were  6  in.,  7  in.,  7J  in.,  8J  in.,  and 

a  diagonal  AB  was  8  in.     What  was  the  scale  of  his  drawing? 


140 


RATIONAL    GRAMMAR    SCHOOL    ARITHMETIC 


FIGURE  79 


7.  On  the  same  board  and  to  the  same  scale 
as  in  problem  6   the  farmer  drew  a  rectangle 
12  in.  by  8  in.  (like  Fig.  79,  but  larger).     What 
was  the  area  of  the  rectangle?    How  many  square 
rods  did  the  rectangle  represent?  how  many  acres? 

8.  Both  blocks   were   then   sawed   out   and 
weighed.     The  irregular  block  weighed  1T\  Ib.  and  the  rectangle 
2J  Ib.     What  was  the  area  of  the  irregular  field? 

It  is  desired  to  find 
the  areas  of  the  outer 
faces  of  the  separate 
stones,  1,  3,  3,  and  so 
on,  of  an  elliptical 
arch.  The  span  of  the 
arch  is  24'  and  the  rise 
9',  as  shown.  A  draw- 
ing of  the  arch  (like 
Fig.  80,  bat  larger)  was  made  on  heavy,  firm  cardboard  (bristol 
board)  to  a  scale  of  1: 12.  The  pieces  of  cardboard  representing 
the  separate  stones  were  then  carefully  cut  out  and  weighed.  A 
square  inch  of  the  cardboard  was  also  cut  oat  and  weighed.  The 
weights,  in  grains,  of  the  several  pieces  are  here  tabulated : 


sq. 
No. 
1 

in. 

WT. 

216 

No. 

7 

-  48  gr. 
WT. 

276 

2 

168 

8 

156 

3 

156 

9 

'  120 

4 

150 

10 

162 

5 

126 

11 

168 

6 

150 

12 

180 

13 

210 

9.  Noticing  that  48  gr.  is  the  weight  of  1  sq.  in.  of  the  card- 
board and  that  it  represents  a  square  foot  on  the  actual  arch ;  find 
from   the  numbers  of   the  table  the  areas  of   the  faces  of  the 
several  stones  forming  the  arch. 

10.  A  block  of  the  stone  1  ft.  square  and  as  long  as  the  hori- 
zontal thickness  of  the  archstones  weighed  688  Ib.     Noticing  that 
the  weight  48  gr.  may  also  be  taken  to  represent  this  688  Ib.,  find 
the  weights  of  the  several  archstones. 


INTRODUCTION   TO    FRACTIONS 


141 


INTRODUCTION  TO  FRACTIONS 

KATIO  AND  PROPORTION 

DEFINITION. — The  ratio  of  one  number  to  another  is  the  quotient  of 
the  first  number  divided  by  the  second. 

1.  What  is  the  ratio  of  A  to  0?  of  0  to  A?  of  B  to  A?  of  A 
to  £?  of  B  to  0?  of  0  to  B?  of  C  to  B?  of  C  to  A?  of  (7  to  0? 
of  0  to  (7?  of  D  to  C?  of  D  to  £?  of  £  to  0?  of  0  to  Z>? 


0 


f 


FIGURE  81 

2.  In  the  second  square,  compare  each  of  the  parts  E,  F,  and 
G  with  0;  compare  0  with  each  of  these  parts. 

3.  Compare  each  division  with  each  of  the  others,  separately. 

4.  In  the  third  square,  J  of  H  is  how  many  times  J?    J  of  I  is 
what  part  of  H*?     \  of  0  is  how  many  times  /? 

5.  What  is  the  ratio  of  3"  to  1"?  of  1"  to  3"?  of  1"  to  J"?  of 
I"  to  1"? 

6.  What  is  the  ratio  of  6  to  0?    of  50  to  50?   of  j  to  J?  of 
a  to  a?  of  x  to  £? 

7.  What  is  the  ratio  of  any  two  equal  numbers? 

,  _  x  _  N          J  is  a  short  way  of  writing  the  following 

'  -  '  -  '     "  L^_,    e  xpr  essio  n  s  : 

f£ 

FIGURE  82  The  ratio  of  1  to  4, 


1  :4, 
i:l- 
1:4  is  read  "1  to  4,"  and  J:l  is  read  "£  to  1." 

8.  What  is  the  ratio  of  1  ft.  to  1  yd.?  of  1  hr.  to  1  da.?  of 


1  nickel  to  1  dime?   of  1  in.  to  12  in.? 
to  1  in.?  of  1  mi.  to  1  mile? 


of  1  ft.  to  3  ft.?   of  12  in. 


142  RATIONAL   GRAMMAR   SCHOOL   ARITHMETIC 

The  ratio  of  a  12-in.  line  to  a  1-in.  line  is  called,  also,  the 
measure  of  the  12-in.  line  by  the  1-in.  line. 

9.  What  is  the  ratio  of  1  sq.  ft.  to  a  3-in.  square 


6 


(see  Fig.  83)?  of  1  sq.  ft.  to  1  square  inch? 

The  ratio  of  a  square  foot  to  a  3-in.  square  is 
the  measure  of  a  square  foot  in  terms  of  the  8-in. 
square. 

Finding  the  ratio  of  the  foot  to  the  inch  is  the  same  as  meas- 
uring the  foot  by  the  inch.  Furthermore,  to  find  the  ratio  of  any 
number,  or  quantity,  to  any  other  number,  or  quantity,  is  to  find 
the  quotient  of  the  first  number,  or  quantity,  divided  by  the 
second. 

The  result  of  measuring  one  number  by  another  is  called  the 
numerical  measure  of  the  first  by  the  second. 

The  ratio  of  one  number  to  another,  or  the  measure  of  one 
number  by  another,  can  be  found  only  when  both  numbers  are 
expressed  in  the  same  unit. 

10.  Measure  the  avoirdupois  pound  by  the  ounce ;  the  foot  by 
the  inch;  the  inch  by  the  foot;  the  ounce  by  the  pound;  the  gal- 
lon by  the  quart ;  the  bushel  by  the  quart. 

11.  Measure  the  rod  by  the  yard;    the  mile  by  the  rod;  the 
square  yard  by  the  square  foot ;  the  quart  by  the  gallon. 

12.  Measure  8  ft.  by  4  ft. ;  1G  ft.  by  64  ft. ;  a  square  mile  by 
a  square  rod;   an   acre   by  a  square  rod;   a  square  mile   by  an 
acre. 

13.  Measure  80  by  8 ;  80  by  4;  3  by  4;  4  by  3;  9  by  18;  18  by 
9;  125  by  25. 

14.  Measure  a  by  #;   a  by  2«;  b  by  a\  %a  by  a\    6x  by  2z;  2# 
by  6z. 

15.  If  5  apples  cost  4$,  what  will  35  apples  cost? 

SOLUTION. — How  many  fives  of  apples  in  35  apples? 

How  much  does  one  five  of  apples  cost? 

How  much  then  do  35  apples,  or  7  fives  of  apples,  cost? 
This  analysis  may  be  thought  of  in  the  form  of  ratios. 

Cost  of  35  apples       x  35       x 

_     .     f    K    F  |     =  77,,  or  more  briefly  -=  =  -^  (1) 

Cost  of   5  apples      4^  5       4f 

and  we  have  to  find  a  number  which  has  such  a  ratio  to  4^  as  35  apples 
has  to  5  apples. 

This  leads  us  to  an  equation  of  ratios. 

DEFINITION.— An  equation  of  ratios  is  called  a  proportion. 


INTRODUCTION   TO    FRACTIONS 


143 


§91.  Proportion. 

In   all   these   problems  use  the  equation   form  of   statement 
like  (1)  in  the  solution  of  problem  15  of  the  preceding  section. 

1.  If  1  doz.  oranges  cost  $0.30,  what  will  4  oranges  cost  at  the 
same  rate? 

2.  If  a  yard  of  cloth  costs  $1.50,  at  the  same  rate  what  will  J  yd. 
cost? 

3.  If  a  3-in.  square  of  tin  costs  ^,  what  will  a  square  foot  cost 
at  the  same  rate? 

4.  If  6  qt.  of  oil  cost  15^,  what  will  3  gal.  cost  at  tho  same 
rate?  (4  qt.  =  1  gal.) 

5.  Two  rectangular  flower  beds  have  the  same  shape.     One  ia 
3  ft.  wide  and  4  ft.  long;  the  other  is  0  ft.  wide.    How  long  is  it? 

6.  Two  books  are  of  the  same  shape  but  of  different  sizes. 
One  is  5"  wide  x  7-J-"  long.     The  other  is  15"  long.     How  wide 
is  it? 

7.  In    two 
triangles  of  the 
same  shape,  like 
those  of  Fig.  84, 

12",  and  I  =  4", 
how  long  is  B? 

8.  In  triangles  of  the  same  shape,  like  those  of  Fig.  84,  (2),  if 
a  =  4",  A  =  12",  and  b  =  5",  how  long  is  B?      If  a  =  6",  b  =  8", 
and  B  =  32",  how  long  is  A?     If  b  =  7",  B  =  35",  and  A  =  30", 
how  long  is  a? 

9.  In  triangles  of  the  same  shape,  as  those  of  Fig.  84,  (3),  if 
A  =  21",  b  =  9",  and  B  =  27",  how  long  is  a?    If  a  =  5£",  A  =  22", 
and  B  «  30",  how  long  is  £? 

10.  In  triangles  of  the  same  shape,  like  those  in  Fig.  84,  (4), 
if  a  =  3",  A  =  9",  and  C=  6",  how  long  is  c?    If  A  =  24",  c  =•  4", 
and  C  =  12",  how  long  is  a? 

§92.  Fractions  as  Ratios  and  as  Equal  Farts. 

1.  Into  how  many  equal  parts  is  A  (Fig.  85)  divided?    What  is 
one  of  the  parts  called? 


(i) 


(4) 


144 


RATIONAL   GRAMMAR   SCHOOL   ARITHMETIC 


2.  Give  the  number  of  equal  parts  and  the  name  of  one  of  the 
parts  in  B>,  C\  D\  E\  F\  0;  H\  /;  /. 

ABODE 


FIGURE  85 

3.  One  part  of  A  equals  how  many  of  the  smallest  parts  of  JB? 
of  (7?  of  D  ?  of  E?  of  G  ?  of  /?  of  /? 

4.  One  part  of  F  equals  how  many  of  the  smallest  parts  of  6r? 
of  H?  of/?   of/?  of  D?  of  EV 

5.  Write  the  fraction  that  names  one  of  the  equal  parts  of  J, 
as  divided  in  the  illustration. 

6.  Measure  A  by  one  of  the  equal  parts  into  which  it  is  divided. 

Ans.  2  halves,  written  f . 

7.  Measure  the  shaded  part  of  B  by  one  of  its  4  equal  parts. 


Ans. 


f,  read  "2  fourths." 


8.  Measure  the  unshaded  part  of  B  by  the  same  unit. 

9.  Using  one  of  the  8  equal  parts  of  C  as  a  unit,  express  the 
shaded  part  of  C  in  this  unit;  the  right  half  of  the  shaded  part. 

10.  Express  E  in  terms  of   the  smallest   unit   it   is  divided 
into ;  the  upper  half  of  E. 

11.  Express  in  terms  of  the  smallest  unit  each  figure  is  divided 
into: 

(1)  the  whole  square  F;  the  shaded  part;  the  unshaded  part. 


(2) 
(3) 
(4) 
(5) 


0; 

H; 

I; 

J; 


DEFINITION.— |  means  3  of  the  i's  (one  fourths)  of  some  number,  or 
measured  quantity.  The  £  is  called  the  fractional  unit.  The  fractional 
unit  of  any  fraction  is  one  of  the  equal  parts  expressed  by  the  fraction. 


INTRODUCTION"   TO    FRACTIONS  145 

12.  What  is  the  fractional  unit  of  each  of  the  following  frac- 
tions: f,  |,  f,  t,  4,  fr,  f,  1,  H,  t>  i,  i,  4, 1,  A,  A,  A,  A? 

13.  What  is  the  fractional  unit  of  a  fraction  having  the  num- 
ber 7  below  the  line?  the  number  6?  9?  12?  15?  25?  18?  11?  64? 

14.  How  many  fractional  units  are  expressed  by  the  1st  frac- 
tion of  question  12?  by  the  2d  fraction?   the  3d?  4th?  5th?  6th? 
7th?     Draw  a  square  and  divide  it  free-hand  by  lines  showing 
the  meaning  of  each  of  the  first  8  fractions  of  problem  12. 

15.  What  does  the  number  written  below  the  line  of  a  fraction 
express? 

16.  What  does  the  number  written  above  the  line  of  a  fraction 
express? 

DEFINITIONS. — The  number  above  the  line  is  called  the  numerator 
(meaning  number er). 

The  number  below  the  line  is  called  the  denominator  (meaning 
namer). 

17.  What  are  the  fractional  units  of  fractions  having  the  fol- 
lowing denominators : 

3?     7?     12?     18?     24?     125?     75?     19?     a?    x? 

18.  Point  out  the  fractions  of  question  12  that  have  the  same 
fractional  units. 

DEFINITION. — The  numerator  and  the  denominator  are  together  called 
the  terms  of  a  fraction. 

19.  If  each  half  of  a  square  is  divided  into  2  equal  parts,  into 
how  many  equal  parts  is  the  whole  square  divided? 

20.  Into  how  many  equal  parts  is  a  whole  divided,  if  each  half 
of  it  is  divided  into  3  equal  parts?  into  4  equal  parts?   into  5?   6? 
7?    8?    9?    10?    11?    25? 

21.  Into  how  many  equal  parts  is  a  whole  divided,  if  each  third 
of  it  is  divided  into  2  equal  parts?  into  3  equal  parts?  into  4?   5? 
6?  7?  8?  9?  10?  12?  15?  20?  30? 

22.  Into  how  many  equal  parts  is  a  whole  divided  by  dividing 
each  of  its  fifths  into  2  equal  parts?  3  equal  parts?  4?    5?   6?   7? 
8?  9?  10?  20?  30? 

Give  solutions  of  such  as  the  following  and  explain : 

23.  Express  as  sixths,  £,  f ,  f ,  J,  }. 

24.  Express  as  twelfths,  J,  J,  |,  J,  f ,  J,  f 


146  RATIONAL   GRAMMAR   SCHOOL   ARITHMETIC 

25.  In  question  12  point  out  pairs  of  fractions  that  are  not 
expressed   in   the   same   fractional    unit,    but   may   easily   be   so 
expressed. 

26.  Express   the   following   pairs   of  fractions   as  equivalent 
fractions  having  the  same  fractional  unit,  or,  what  is  the  same 
thing,  having  a  common  denominator: 

I  and  f;  |  and  |;  f  and  f ;  f  and  f ;  f  and  | ;  J  and  T5T. 


COMMON   FRACTIONS 

§93.  To  Reduce  Fractions  to  Higher,  Lower,  and  Lowest  Terms. 

_  %  _  1.  Express  £  as  4ths;  8  ths; 

I    .    1.1,1.1,1,1,1,1     IGths. 

2.  Express  J  as  Gths  ;  9ths  ; 


* 


i_J     .     L_._J     .     i     im     12ths;  18ths. 

^  ^  3.  Express   as   3ds   f;    |; 

H;  H;  i«;  If- 


4.  Express  as  7ths  ^ ;  f  f ; 
FIGURE  86  24.54 

?6>     SS* 

5.  Compare  the  values  of  these  fractions  after  reducing  each 
to  its  lowest  terms  (that  is,  to  the  smallest  possible  whole  num- 
bers for  numerators  and  denominators) . 

i;  1;  A;  H;  W;  ifo;  ttt- 

6.  Express  f  as  lOths;  15ths;  20ths;  35ths;  75ths;  lOOths. 

7.  By  what  must  you  multiply  the  numerator  of  f  to  make 
the  numerator  the  same  as  that  of  |?    By  what  must  you  multiply 
the  denominator  of  f  to  make  the  denominator  the  same  as  that 
of  |?     Draw  a  square  and  divide  it  free-hand  to  show  the  com- 
parative sizes  of  f  and  |   (see  square  (7,  Fig.  85,  §92). 

8.  By  what  must  you  multiply  each  term  of  J  to  obtain  T\? 
f  i?  It?  TOSO?  f  U?    What,  then,  is  the  value  of  each  fraction  of 
problem  5? 

9.  How  then  may  the  value  of  a  fraction   be  expressed  in 
higher  (or  larger)  terms? 


COMMON   FRACTIONS  147 

A  fraction  may  be  expressed  in  higher  terms,  without  chang- 
ing its  value,  by  multiplying  both  terms  by  the  same  number. 

10.  Express  the  following  fractions  in  higher  terms: 

!;!;!;  if;  if;  A;  IJ- 

11.  How  can  you  obtain  the  terms  of  f  from  the  terms  of  f  ? 

of  A?  offi?  ofTy0? 

12.  What  fraction  do  you  obtain  by  dividing  each  term  of 
£s  by  4?     What,  then,  is  the  relation  as  to  value  between  the 
fractions  /8  and  4? 

13.  By  using  a  rectangle,  divided  as  in  Fig. 
87,  show  that  fa  of  the  rectangle  equals  \  of  it. 

14.  In  what  lower  terms  can  f  f  be  expressed?  FIGURE  s? 
iVo?   Ji?     Show   that   f  £  =  f   by  means  of  a  properly  divided 
rectangle. 

15.  What  effect   on   the  value  of  a  fraction  is  produced  by 
dividing  each  of  its  terms  by  the  same  number? 

16.  How  may  the  value  of  a  fraction  be  expressed  in  lower 
terms? 

PRINCIPLE  I. — Multiplying  or  dividing  both  terms  of  a  frac- 
tion by  lite  same  number  changes  tlte  form  of  the  fraction  without 
altering  its  value. 

Fractions  are  most  easily  used  when  their  numerators  and 
their  denominators  are  the  smallest  possible  whole  numbers. 

DEFINITION. — The  fractions  are  then  said  to  be  in  their  lowest  terms. 
Among  the  many  ways  in  which  J-f  can  be  written  are  these: 

H;  T"*;  t;  I;  f 

17.  Show  how  J  is  obtained  from  £f  by  the  use  of  Principle  I. 

18.  Give  the  values  of  these  fractions  in  their  lowest  terms: 

5    .     6.     21.      8.     35.     32 

TO-;  f»  fi;  TV;  HJ  M- 

To  obtain  the  value  of  a  fraction  in  the  smallest  possible  terms, 
it  is  necessary  to  divide  both  terms  by  the  largest  exact  common 
divisor  of  both  terms. 

DEFINITION. — This  divisor  is  called  the  greatest  common  divisor  of  the 
terms.  It  is  indicated  by  the  initial  letters  G.  C.  D. 


148  RATIONAL    GRAMMAR   SCHOOL    ARITHMETIC 

Dividing  both  terms  of  a  fraction  by  their  G.  0.  D.  reduces 
the  fraction  to  its  lowest  terms. 

19.  These  fractions  are  in  their  lowest  terms: 

i;  i;  t;  i;  I;  f;  fl;  if;  if. 

Among  these  fractions  is  there  a  factor  common  to  any  numer- 
ator and  its  denominator? 

DEFINITION.  —  Two  numbers  that  have  no  common  factor,  except  1, 
are  said  to  be  prime  to  each  other. 

20.  How  then  may  we  test  by  means  of  factors  whether  a 
fraction  is  in  its  lowest  terms? 

21.  Reduce  these  fractions  to  their  lowest  terms: 

16.     27.     60.     25.     39.       74    .     169 
¥4>     8T>    THFJ    T^J     5  ¥  >    TIT  5    3i5T 

Tho  problem  of  finding  the  common  divisors  of  such  numbers 
as  are  in  the  numerators  and  the  denominators  of  the  last  two 
fractions  is  tedious  without  a  general  method  of  finding  the 
G.  C.  D.  of  numbers. 

§94.  Factors,  Prime  and  Composite. 

NOTE.  —  Review  tests  of  divisibility,  §63,  pp.  85,  86,  and  use  them 
through  this  section  and  the  next,  when  searching  for  factors. 

1.  Write  down  all  the  factors,  or  exact  divisors,  of  36  and 
of  48. 

DEFINITION.  —  By  factor  is  here  meant  exact  divisor,  or  a  divisor 
that  is  contained  without  a  remainder. 

CONVENIENT  FORM 

2)36,  48 

2)18   24  The  factors  of  36  are  1,  2,  3,  4,  6,  9,  12, 

-T  18,  and  36. 

The  factors  of  48  are  1,  2,  3,  4,  6,  8,  12, 


2)3>    4  16,  24,  and  48. 

3,     2 

2.  What  factors  are  common  to  both  36  and  48? 

3.  What  is  the  greatest  factor  that  is  common  to  36  and  48? 

4.  Arrange  the  factors  of  42  and  105  as  the  factors  of  36  and 
48  are  arranged  in  problem  1,  and  answer  questions  like  2  and  3 
for  the  factors  of  42  and  105. 


COMMON    FRACTIONS  149 

5.  Give  the  factors  that  are  common  to  the  3  numbers,  18,  24, 
and  36.     Give  the  greatest  factor  common  to  all  three  numbers. 

6.  Answer  questions  like  those  of  problem  5  for  these  numbers: 

(1)  12,  72,  and  84;    (3)  42,  98,  and  168;   (5)  36,  84,  96,  and  108; 

(2)  27,  36,  and  81;  (4)  32,  48,  and  96;  (6)  44,  99,  110,  and  121. 

7.  Are  there  factors  of  5  other  than  itself  and  1?    of  7?  of  13? 
of  17?  of  23?  of  29?  of  31?  of  37? 

DEFINITIONS.— A  number  that  has  no  [factors  except  itself  and  1  is  a 
prime  number.  A  number  that  has  factors  beside  itself  and  1  is  a  com- 
posite number. 

8.  Name  the  prime  numbers  from  1  to  100;    the  composite 
numbers  from  1  to  100. 

QUERY. — Is  2  a  prime  or  a  composite  number? 

DEFINITIONS. — Any  number  that  can  be  exactly  divided  by  2  is  an 
even  number.  All  other  whole  numbers  are  odd  numbers. 

9.  How  can  an  even  number  be  quickly  recognized?     (See  test 
of  divisibility  by  2,  p.  85.) 

10.  Name  the  even  numbers  from  0  to  50;    the  odd  numbers 
from  1  to  50. 

11.  Tell  what   numbers  of   these  are   (1)    even,   (2)  odd,  (3) 
prime,   (4)  composite: 

5,  8,  9,  2,  21,  15,  19,  26,  27,  38,  41,  42. 

12.  Mention  some  numbers  that  are  both  odd  and  composite; 
even  and  composite;  odd  and  prime. 

13.  Factors  of  28  are  7  and  4.     As  7  is  a  prime  number  it  is 
called  a  prime  factor.     What  is  a  prime  factor? 

14.  Write  the  prime  factors  of  21;  of  24;  of  25;  of  27;  of  30; 
of  32;  of  36;  of  37. 

NOTE.— Write  out  the  prime  factors  of  these  numbers  as  they  are 
here  written  out  for  96 : 

96  =  2X2X2X2X2X3. 

15.  How  many  times  does  2  occur  as  a  prime  factor  in  96? 
This  may  be  indicated  by  writing  96  thus:  26  x  3.     The  small  5 
written  to  the  right  and  above  the  2  is  to  show  how  many  times  2 
is  to  be  used  as  a  factor. 


150  RATIONAL    GRAMMAR    SCHOOL   ARITHMETIC 

§95.  Greatest  Common  Divisor  by  Prime  Factors. 

1.  What  are  the  prime  factors  of  36  and  48?  (See  problem  1, 
§94.)     Write  36  in  the  form  given  for  96   in  problem  15  of  the 
last  section.     Write  48  also  in  this  form. 

2.  Will  each  of  the  common  prime  factors  of  36  and  48  divide 
the  G.  C.  D.  of  36  and  48?     Will  the  product  of  all  the  common 
prime  factors  divide  the  G.  C.  D.  exactly? 

3.  Answer  questions  like  1  and  2  for  the  numbers  42  and  105 ; 
for  96  and  216;  for  75  and  250. 

4.  When  any  common  prime  factor  occurs  repeatedly  in  one  or 
both  of  the  numbers,  how  often  does  it  occur  in  the  G.  C.  D.  of 
those  numbers?     Answer  by  examining  these  pairs  of  numbers : 

(1)  12  and  48;  (3)  75  and  250; 

(2)  54  and  405;          (4)   98  and  343. 

5.  Make  a  rule  for  finding  the  G.  C.  D.  of  two  or  more  numbers 
from  their  common  prime  factors  when  no  common  prime  factors 
are  repeated  in  any  of  the  numbers.     Test  your  rule  by  finding 
the  G.  C.  D.  of  30  and  42;  of  105  and  231 ;  of  30  and  70. 

6.  Make  a  rule  that  will  give  the  G.  C.  D.  of  two  numbers 
when  one  or  more  of  the  common  prime  factors  is  repeated  in 
one  or  both  of  the  numbers,  and  test  your  rule  by  finding  the 
G.  C.  D.  of  72  and  108;  of  288  and  648;  of  675  and  1125. 

7.  Find  the  G.  G.  D.  of  792  and  1080. 

CONVENIENT  FORM 
2)    792         1080 

2)  396  540  793  =  23  X  32  X  11 

2)198  270  1Q80  =  23  X  33  X    5 

3)99  135"  G.  C.  D.  =  23  X  8-'  =  8  X  9  =  72 

3)33 45_ 

11          3)15 
~5~ 

PRINCIPLE  II. — The  G.  0.  D.  of  two  or  more  numbers  equals  the 
product  of  all  the  prime  factors  common  to  all  the  numbers  ^  each 
common  prime  factor  being  used  the  smallest  number  of  times  it 
occurs  in  any  one  of  the  given  numbers. 

8.  Find  the  G.  C.  D.  of  the  sets  of  numbers  in  problem  6,  §94. 


COMMON   FRACTIONS 


151 


The  method  of  finding  the  G.  C.  D.  by  factors  is  tedious  when 
the  numbers  are  large  and  the  factors  are  not  readily  detected. 
We  now  give  a  method  for  large  numbers. 


§96.  Greatest  Common  Divisor  by  Successive  Division. 

To  obtain  quickly  the  G.  C.  D.  of  large  numbers,  the  following 
method  is  useful  : 

1.  Find  the  G.  C.  D.  of  851  and  10,952. 

CONVENIENT  FORM 

851)10952(12 


851 

2442 

1702 


llf>740(6 
666 

"74)111(1 
_74_ 

37)74(2 
74 
0 

The  last  divisor,  37, 
which  gives  the  remain- 
der 0,  is  the  G.  C.  D. 


2.  Find  the  G.  C.  D.  of  these  numbers: 


This  is  called  the  method  of  finding 
^6  ®m  ®'  ^*  ^7  successive  division. 

To  find  the  G.  C.  D.  of  more  than  two 
numbers,  first  find  the  G.  C.  D.  of  two  of 
the  numbers,  then  the  G.  C.  D.  of  this 
G.  C.  D.  and  a  third  number,  and  so  on. 


(1)  1413  and  4710; 

(2)  432  and  5184; 

(3)  14,457  and  27,450; 


(4)  247,  969,  and  1235; 

(5)  272,  357,  and  425; 

(6)  517,  752  and  1034. 


3.  Eeduce  these  fractions  to  their  lowest  terms  by  finding  the 
G.  G.  D.  of  the  numerator  and  the  denominator,  and  dividing 
both  terms  by  the  G.  C.  D.  : 

(i)W;     (3)  MM;     (*)  VWi;     (?)  Wt; 
(4)  tttt;     (6)  K«;     (8) 


NOTE.  —  In  reducing  fractions  to  their  lowest  terms  it  is  best  to  reject 
(divide  out)  at  once  from  both  numerator  and  denominator,  any  common 
factors  that  can  be  readily  seen,  before  seeking  the  G.  C.  D.  of  the 
terms.  This  will  often  be  advantageous,  if  the  tests  for  divisibility. 
§63,  pp.  85,  86,  are  well  known. 


152  RATIONAL   GRAMMAR    SCHOOL   ARITHMETIC 

§97.  Greatest  Common  Divisor  of  Lines  with  Compass. 

PROBLEM.— Find  the  G.  C.  D.  of  the  lines  AB  and  CD. 

E F  K   R          EXPLANATION. — Take  the  distance 

A '    CD  between  the  compass  points,  and 
«      G    H     I    ~  placing  the  pin-foot  on   A,  mark  off 

AE=  CD.     Then  mark  off  EF=  CD. 

P,T_.  This  leaves  the  remainder  FB,  which 

is  less  than  CD.    This  would  be  writ- 
ten FB  <  CD,  and  read  "FB  is  less    than  CD." 

DEFINITION.— The  sign  <  is  called  the  sign  of  inequality.  The  point 
of  the  <  is  always  turned  toward  the  smaller  number. 

Thus,  FB  <  CD  means  that  FB  is  less  than  CD, 

but  CD  >  FB  means  that  CD  is  greater  than  FB. 

Now  take  the  distance  FB  between  the  compass  feet,  and  mark 
off  on  CD  the  spaces  CG,  GH,  and  HI,  each  equal  to  FB,  until 
a  remainder  ID  is  left,  such  that  ID  <  FB. 

Then  ID  is  found  to  be  contained  just  twice  in  the  former 
remainder  FB.  ID,  the  last  divisor  or  measure,  is  then  the  greatest 
common  measure,  or  divisor,  of  both  AB  and  CD. 

To  see  that  it  is  a  common  measure  answer  these  questions : 

1.  How  many  times  is  ID  contained  in  FB? 

2.  How  many  times  is  FB  contained  in   CI? 

3.  How  many  times  is  ID  contained  in  £77?  in  CD? 

4.  How  many  times  is  CD  contained  in  AF? 

5.  How  many  times  is  ID  contained  in  AF?  in  AB? 
Consequently  ID  is  a  common  measure  of  AB  and  CD. 

To  see  that  any  common  measure  of  AB  and  CD  must  measure 
ID  exactly,  answer  these  questions : 

6.  If  any  line,  as  ID,  is  contained  7  times  in   CD,  how  many 
times  is  it  contained  in  AF? 

7.  If  ID  is  contained,  say  16  times,  in  AB  and  14  times  in 

AF,  how  often  must  it  be  contained  in  FB? 

• 

In  the  same  way  it  could  be  seen  that  each  remainder  after  a 
number  of  applications  of  the  preceding  remainder  would  exactly 
divide  this  last  remainder. 

8.  Draw  two  lines  of  different  lengths  on  the  blackboard,  and 
with  crayon  and  string  find  their  greatest  common  measure  or 
divisor. 


COMMON    FRACTIONS  153 


?98.  Problems. 


In  each  case  when  the  answer  to  the  problem  is  a  fraction,  or 
ratio,  it  must  be  expressed  in  its  lowest  terms. 

1.  A  man  works  48  da.  out  of  64  da.     What  part  of  64  da. 
does  he  work? 

2.  A  grocer  bought  24  boxes  of  oranges  and  sold  16.     What 
part  of  his  purchase  was  sold? 

3.  Out  of  56  bu.  of  potatoes  a  huckster  sold  49  bu.     What 
part  of  his  potatoes  was  sold? 

4.  A  street  car  on  one  line  makes  a  weekly  average  of  72  trips. 
A  car  on  another  line  makes  an  average  of  144  trips.     What  is 
the  ratio  of  the  former  to  the  latter? 

5.  A  man  pays  $50  for  wood  and  $125  for  coal  in  one  season. 
Find  the  ratio  of  the  cost  of  the  wood  to  the  cost  of  the  coal. 

6.  Out  of  1000  ft.  of  lumber  purchased,  500  ft.  were  used  for 
the  flooring  of  two  rooms.     Let  x  equal  the  ratio  of  the  number  of 
feet  of  lumber  used  to  the  total  number  of  feet  purchased.     Find 
the  value  of  x. 

7.  Out  of  126  bu.  of  oats,  a  livery  man  fed  63  bu.  in  1  wk. 
Let  y  equal  the  ratio  of  the  quantity  of  oats  bought  to  the  quan- 
tity of  oats  fed.     Find  y. 

8.  In  667  Ib.  of  sandy  loam  there  were  377  Ib.  of  sand  and 
gravel.     What  part  of  the  soil  by  weight  was  sand  and  gravel? 

9.  The  human  body  needs  about  94.5  oz.  of  water  and  solid 
food  each  day,  of  which  64.8  oz.  should  be  water.     The  water  is 
what  part,  by  weight,  of  the  total  quantity  of  solid  food  and  water? 

SUGGESTION. — Both  terms  of  the  fraction  may  be  multiplied  by  10 
without  changing  the  value  of  the  fraction.  Then  find  the  G.  C.  D.  of 
the  new  terms  and  divide  both  new  terms  by  it. 

10.  The  total  population  of  a  certain  city  is  36,729,  of  which 
12,243  are  colored  and  10,017  are  foreign.    What  part  of  the  entire 
population  is  colored?     What  part  is  foreign? 

11.  What  part  of  the  total  population  is  made  up  of  colored 
and  of  foreign  persons? 

The  solution  of  problem  10  will  show  that  J  of  the  population 
is  colored,  and  that  T3T  of  it  is  foreign.  We  can  solve  problem  11 
if  we  can  find  the  value  of  J  +  -fa. 


154  RATIONAL    GRAMMAR   SCHOOL    ARITHMETIC 

12.   A  boy  spent  f  of  his  money  and  gave  away  £  of  it;  what 
part  of  it  did  he  have  left? 

To  solve  this  problem  we  must  know  how  to  add  and  subtract 
fractions.     This  is  what  we  shall  study  next. 

§99.  Fractions   Having   a    Common    Fractional   Unit   (a   Common 
Denominator). 

1.  Complete  these  equations: 

fyd.+fyd.  =  !gal.+fgal.=  £  +  £  = 

|  hr.  +  J  hr.    =  T6-g-  -f  T97  = 

f  wk.  +  f  wk.  =          y\  +  y1^  = 
f  bu.  +ibu.  =  ^  +  ^    = 

NOTE.  —  -^  is  read,  "5  divided  by  a,"  and  -^  is  read,  "x  divided  by  z." 
!~  is  read,  "a  plus  b  divided  by  c." 

2.  Make  a  rule  for  adding  fractions  having  a  common  denom- 
inator. 

3.  Complete  these  equations: 

I  - 1  =  T\  ~  Tsy  =         if  -  T3s  =  I!  nr-  -  &  hr.  = 


NOTE. — ^-  is  read,  "a  minus  Z>  divided  by  c." 

4.  Make  a  rule  for  finding  the  difference  between  two  fractions 
having  a  common  denominator. 

5.  Denoting  any  two  fractions  having  common  denominators 
by  j  and  •§•»  state  in  symbols : 

PRINCIPLE  III. — The  sum,  or  the  difference,  of  any  two  frac- 
tions having  common  denominators  equals  the  sum,  or  the  difference, 
of  their  numerators,  divided  by  the  common  denominator. 

§100.  Fractions  Easily  Reduced  to  Common  Fractional  Unit. 

1.  i  equals  how  many  12ths?  8ths?  IGths?  20ths?  64ths? 
lOOths? 


COMMON    FRACTIONS 


155 


2.  T\  hr.  +  i  hr.  =  x  hr.     What  is  the  value  of  x? 
SOLUTION.—  £\  hr.  +  T\  hr.  =  {°  hr.  =  f  hr.    x  =  f. 

3.  TV  br.  -  i  hr.  =  y  hr.     What  is  y? 

4.  Find  these  sums  and  differences: 


t+t  -  I  -A- 


A 


A+A-A 


§101.  Growth  of  Trees. 

Twigs,  broken  from  a  scrub  oak,  were  taken  to  the  classroom 
for  measurement.  Some  were  taken  from  the  lower,  some  from 
the  middle,  and  some  from  the  upper  branches.  The  twigs  from 
the  lower  branches  were  grouped  into  three  bunches:  one  bunch 
containing  those  from  the  north  side  of  the  tree;  one,  those  from 
the  east  and  west  sides;  and  one,  those  from  the  south  side.  The 
twigs  from  the  middle  branches  and  also  those  from  the  top 
branches  were  treated  in  the  same  way.  Measures  were  made 
from  the  tips  of  the  twigs  to  the  first  cluster  of  rings  marking 
the  year's  growth.  In  the  table  below,  they  are  given  in  inches 
to  the  nearest  eighth  of  an  inch. 

Similar  measures  were  made  of  the  distances  between  the 
first  and  the  second  clusters  of  rings,  then  between  the  second  and 
the  third  clusters,  and  finally  between  the  third  and  the  fourth 
clusters.  Pupils  are  urged  to  make  such  measures  and  to  tabu- 
late and  study  them  as  is  done  in  the  table  and  problems  below. 

GROWTH  OF  OAK  TWIGS  FOR  LAST  YEAR,  MEASURED  IN  INCHES 


LOWER  BRANCHES 

MIDDLE  BRANCHES 

TOP  BRANCHES 

N. 

E.&W. 

s. 

N. 

E.&W. 

's. 

N. 

E.&W. 

s. 

5* 

<H 
41 
5] 
64 

6f 
43 

<H 

6* 
6^ 

<H 

6! 

<n 

8f 

^ 

11 

7J 
8! 

5£ 

n 

5* 

n 
5 

•  a 

1 
?! 

H 

34 
21 

af 
5| 

2f 
21 
2| 
4| 

3£ 

4| 

3| 
3| 

2? 
4* 

Sum* 
Averages 

Lateral  Growth 

Terminal 
Growth  — 

i 

156  RATIONAL   GRAMMAR   SCHOOL   ARITHMETIC 

1.  Find  from  the  table  the  average  growth  of  the  twigs  from 
the  lower  branches  on  the  north  side  of  the  tree;  from  the  east 
and  west  sides;  from  the  south  side. 

NOTE. — The  average  growth  is  the  sum  of  all  measured  growths  divided 
by  the  number  of  measures. 

2.  Find  the  average  growth  of  twigs  from  the  middle  branches 
for  each  of  the  3  columns ;  from  the  top  branches. 

3.  Where  is  the  average  growth  the  greatest?     How  much  does 
the  greatest  growth  exceed  the  least? 

4.  What  is  the  difference  between  the  average  growth  from  all 
the  middle  branches  and  the  average  growth  from  all  the  lower 
branches?     (Find  the  average  of  the  3  averages). 

5.  The  average  growth  from  both  lower  and  middle  branches 
is  called  the  lateral  growth  of  the  tree.     The  average  from  the 
top  branches  is  called  the  terminal  growth.     For  the  oak,  whose 
measures  are  given  in  the  table,  which  is  the  greater,  the  lateral 
or  the  terminal  growth?  how  much  greater? 

6.  Can  you  account  for  the  differences  of  growth  as  you  find 
them? 

7.  Find  the  average  growth  for  the  entire  tree  by  averaging 
the  averages  for  all  nine  columns. 

8.  Make  a  similar  study  of  the  growth  of  twigs  from  the  ash, 
willow,  poplar,  birch,  hickory,  or  any  other  tree  or  trees  native  to 
your  neighborhood. 

NOTE. — The  next  4  problems  may  be  omitted  if  thought  best  by  the 
teacher: 

9.  Make  measures  on  trees  surrounded  by  other  trees  and  on  trees 
standing  in  isolated  places.    In  which  case  is  the  average  growth  greater? 
how  much  greater? 

10.  Make  measures  on  twigs  of  the  same  kind  of  tree  in  different  soils. 
Is  the"yearly  growth  greater  in  sandy  soils  or  in  loam?  how  much? 

11.  A  short  distance  back  of  the  first  cluster  of  rings  will  be  found 
a  second  cluster,  back  of  this  a  third,  and  so  on.     These  rings  limit  the 
growth  for  last  year,  year  before  last,  and  so  on.     Compare  the  average 
growth   for  last  year  with  the  average  growth    for  year  before  last. 
Which  is  the  greater?  how  much  greater? 

12.  Compare  the  growth  for  last  year  with  the  growth  for  each  of 
the  preceding  three  years.    What  do  you  find?    Can  you  account  for  what 
you  find? 


COMMON    FRACTIONS 


157 


The  following  measures  were  actually  obtained  by  a  class. 
They  are  averages  of  from  10  to  15  different  measures  such  as 
are  tabulated  above,  the  twigs  coming  from  a  healthy  tree  in  each 
case: 


LATERAL  GROWTH,  IN  INCHES 

TERMINAL  GROWTH,  IN  INCHES 

TREE 

Last 
Year 

Previ- 
ous 
Year 

Next 
Previ- 
ous 
Year 

Average 

Last 
Year 

Previ- 
ous 
Year 

Next 
Previ- 
ous 
Year 

Average 

Oak  

4 

4f 

4i 

3A 

2*1 

3| 

Soft  maple.  . 

?t 

9 

4i 

8f 

H 

1 

Birch  

41 

51* 

4* 

8* 

n 

8 

Poplar  

6 

«i 

^ 

2 

n 

2 

13.  Find  the  average  terminal   growth   of   the   oak   for  the 
last  3  yr. ;  of  the  soft  maple ;  of  the  birch ;  of  the  poplar. 

14.  Find   the   average   lateral  growth   for  each  of   the   four 
trees. 

15.  How  much  does  the  terminal  growth  for  each  year  exceed, 
or  fall  short  of,  the  average  terminal  growth,  for  each  tree? 

16.  Answer  the  same  question  for  the  lateral  growth. 

17.  How  much  does  the  terminal  growth  of  the  oak  exceed,  or 
fall  short  of,  the  lateral  growth,  for  each  year? 

18.  Make  and  answer  similar  questions  based  on  the  table. 

19.  From  your  own  measures  make  a  table  similar  to  this  and 
answer  questions  13  to  17  from  your  figures. 

Fractions,  whose  fractional  units  are  not  the  same,  must  be 
expressed  in  a  common  unit  before  they  can  be  added  or  sub- 
tracted. To  do  this  we  must  find  a  fractional  unit,  whose  denom- 
inator can  be  exactly  divided  by  each  of  the  given  denominators. 
It  will  be  the  simplest  always  to  select  the  fractional  unit  whose 
denominator  is  the  least  number  that  each  of  the  given  numbers 
will  divide. 

DEFINITION. — A  denominator  which  is  common  to  two  or  more  frac- 
tions is  called  a  common  denominator.  When  this  common  denominator 
is  the  least  number  that  can  be  found  which  may  be  used  as  a  common 
denominator  of  the  fractions,  it  is  called  the  least  common  denominator, 
and  is  written  L.  C.  D. 

Before  studying  more  difficult  fractions  we  must  learn  how  to 
find  least  common  denominators.  We  begin  by  a  study  of  multi- 
ples, common  multiples,  and  least  common  multiples  of  numbers. 


158  RATIOHAL   GRAMMAR   SCHOOL   ARITHMETIC 

§102.  Multiples. 

ORAL   WORK 

42  yd.  is  exactly  measured  by  7  yd.,  3  yd.,  and  2  yards. 

$21  is  exactly  measured  by  $7  and  $3. 

56  pk.  is  exactly  measured  or  divisible  by  7  pk.  and  8  pecks. 

42  is  a  multiple  of  7,  3,  and  2.  21  is  a  multiple  of  7  and  3. 
56  is  a  multiple  of  7  and  8.  What  are  some  of  the  multiples  of 
3  and  5 ;  of  5  and  7 ;  of  2  and  11? 

1.  What  then  is  a  multiple  of  a  number? 

DEFINITION. — A  number  that  can  be  exactly  divided  by  another 
number,  is  called  a  multiple  of  the  latter  number. 

Following  is  a  list  of  multiples  of  2,  from  2  to  36,  inclusive: 

2,  4,  6,  8,  10,  12,  14,  16,  18,  20,  22,  24,  26,  28,  30,  32,  34,  36. 
Following  is  a  list  of  all  the  multiples  of  3,  to  36 : 

3,  6,  9,  12,  15,  18,  21,  24,  27,  30,  33,  36. 

2.  Why  is  each  number  in  the  first  row  a  multiple  of  2? 

3.  Passing  along  the  rows  underscore  the  numbers  that  are  the 
same  in  both  rows. 

4.  In  what  way  are  these  multiples  of  2  and  3  different  from 
the  rest?  Am.  They  occur  in  both  rows. 

5.  What  is  a  common  multiple  of  two  numbers? 

DEFINITION. — A  number  that  can  be  exactly  divided  by  two  or  more 
numbers,  is  called  a  common  multiple  of  those  numbers. 

6.  Eewriting  the  common  multiples  of  2  and  3  we  have: 

6,  12,  18,  24,  30,  36. 

Can  you  supply  a  few  more  common  multiples  here  without 
extending  the  rows  above  problem  2? 

7.  What  number,  besides  2  and  3,  will  exactly  divide  all  the 
common  multiples  of  2  and  3? 

8.  On  the  blackboard  write  out  rows  of  multiples  of  3  and  of  5, 
like  those  above,  underscoring,  or  writing  in  colored  chalk,  the 
multiples  that  are  common  to  both  rows. 

9.  Will  the  least  of  the  common  multiples  of  3  and  5  divide 
all  the  other   common  multiples?      What   is   the   least  common 
multiple  of  two  numbers? 


COMMON    FRACTIONS  159 

DEFINITION.— The  least  common  multiple  of  two  or  more  numbers  is 
the  least  number  that  is  exactly  divisible  by  each  of  the  numbers.  It 
is  usually  written  L.  C.  M.  for  brevity 

§103.  Finding  the  L.  C.  M. 

WRITTEN   WORK 

1.  Make  lists  of  the  multiples  of  2,  5,  and  7,  and  find  their 
L.  C.  M. 

2.  In  the  same  manner,  find  the  L.  C.  M.  of  3,  4,  and  5. 

3.  Note  that  all  the  given  numbers  in  problems  1  or  2  are 
prime  to  each  other.     In  such  a  case  how  can  the  L.  C.  M.  be 
found  from  the  numbers? 

Tlie  L.  C.  M.  of  two  or  more  numbers,  all  prime  to  each  other, 
is  their  product. 

4.  Illustrate  the  truth  of  this  statement  by  two  sets  of  num- 
bers of  your  own  selection,  one  to  contain  two  numbers,  and  the 
other  three,  the  numbers  of  each  set  being  prime  to  each  other. 

5.  4,  8,  12,  10,  20,  24  are  multiples  of  4;  and  G,  12,  18,  24,  30 
are  multiples  of  6.    What  is  the  least  common  multiple  of  4  and  6? 

G.  This  number  is  the  product  of  4  and  6  divided  by  what? 
2  is  the  common  factor  of  4  and  6. 

7.  Are  4  and  10  prime  to  each  other?     What  is  their  greatest 
common  factor  or  divisor? 

8.  4,  8,  12,  16,  20,  24,  28,  32,  36,  40  are  multiples  of  4,  and 
10,  20,  30,  40  are  multiples  of  10.     What  are  some  common  mul- 
tiples of  4  and  10?     What  is  their  least  common  multiple? 

9.  Divide  the  product  of  4  and  10  by  2,  their  G.  C.  D.         hat 
is   the   result?      Compare  this   result   with   the   last  answer  to 
problem  8. 

The  L.  C.  M.  of  two  numbers  not  prime  to  each  other  is  their 
product  divided  by  their-  greatest  common  divisor  (G.  C.  D.). 

10.  What  is  the  greatest  common  factor  of  15  and  20?     What 
is  the  product  of  15  and  20?     How  can  you  find  the  L.  C.  M.? 

11.  In  the  same  manner  find  the  L.  C.  M.  of  8  and  10;  of  12 
and  24;  of  15  and  25;  of  24  and  GO;  of  40  and  36. 


160  RATIONAL    GRAMMAR    SCHOOL    ARITHMETIC 

§104.  Shorter  Process  for  Three  or  More  Numbers 

ORAL   WORK 

1.  In  a  certain  number  there  are  8  24's;  how  many  12's  are 
there  in  the  number?  how  many  8's?  how  many  6's? 

2.  In  a  certain  number,  #,  there  are  15  6's.     How  many  3's 
are  there  in  x?  how  many  2's? 

3.  If  6  exactly  divides  a  certain  number,  #,  what  other  num- 
bers also  exactly  divide  x? 

4.  If  any  composite  number  exactly  divides  a  number,  ?/,  what 
other  numbers  also  divide  y! 

Any  multiple  of  a  number  is  divisible  by  all  the  prime  factors 
of  that  number. 

WRITTEN    WORK 

1.  Find  the  L.  C.  M.  of  150,  504,  and  540. 

SOLUTION.—         150  =  2x3x5x5  =  2  x  3  X  53. 

504  =  2X2X2X3X3X7  =  23X32X  7. 
540  =  2  X  2  X  3  X  3  X  3  X  5  =  22  X  33  X  5. 

The  L.  0.  M.  is  a  multiple  of  each  of  these  numbers  severally. 
By  the  above  principle  it  must  then  contain  the  prime  factors 
2,  3,  5,  and  7.  But,  since  25  =  52  is  a  factor  of  150  the  L.  C.  M. 
must  contain  5  twice  as  a  Jt'actor.  Since  8,  or  23,  is  a  factor  of 
504,  the  L.  C.  M.  must  contain  2  as  a  factor  three  times.  How 
many  times  must  the  L.  C.  M.  contain  3  as  a  factor?  How  many 
times  must  the  L.  C.  M.  contain  7  as  a  iactor?  The  number 
that  contains  just  these  factors  and  no  others  is  2x2x2x3x 
3x3x5x5x7  =  23x33x52x7  =  37,800. 

37,800  is  the  least  common  multiple,  because  it  contains  the 
necessary  factors  and  no  others.  Any  other  common  multiple 
must  contain  all  these  factors  and  some  other,  and  it  would  there- 
fore be  larger  than  37,800. 

What  prime  factors  must  be  in  the  L.  C.  M.  of  three  numbers? 

If  a  prime  factor  is  repeated  in  one  or  more  of  the  numbers, 
how  often  must  it  occur  in  the  L.  C.  M.? 

PRINCIPLE  IV. — The  L.  C.  M.  of  two  or  more  numbers  is  the 
product  of  all  the  different  prime  factors  of  all  the  numbers,  each 
factor  occurring  the  greatest  number  of  times  it  occurs  in  any  one 
of  the  numbers. 


COMMON    FRACTIONS  161 

2.  Find  the  L.  C.  M.  of  these  numbers  and  show  your  work: 

(1)  9,  12,  18;  (4)  15,  25,  42,  50; 

(2)  6,  24,  30;  (5)     7,  28,  24,  42; 

(3)  18,  30,  36;  (6)     6,  18,  27,  36. 

3.  The  work  can  be   shortened  by  this  arrangement.      Let 

us  solve  problem  2,  (6). 

EXPLANATION. — Select  any  prime 

CONVENIENT  FORM  number,  as  3,  that  will  divide  two 

3)  6,     18,     27,     36  or  more  of  the  numbers  whose  L.  C. 

orV F n1 — To  M.  is  sought.     Divide  it  into  all  the 

^'_^! LI ii — —  numbers  that  are  exactly  divisible 

3)   1,       3,      9,      6  by  it,  writing  in  the  line  below  the 

1,       1,      8,       2  quotients  and  any  numbers  the  chosen 

T    n  M       q  v  9  v  q  v  q  v  9  —  1  n«      divisor  does  not  exactly  divide. 

Select  another  prime  number  and 

do  as  before.  Continue  until  the  numbers  last  brought  down  are  all 
prime.  The  continued  product  of  the  divisors  and  the  numbers  remain- 
ing in  the  last  horizontal  line  is  the  L.  C.  M. 

A  little  study  will  show  this  to  be  merely  a  more  convenient  way  of 
finding  the  factors  described  in  Principle  IV. 

The  least  common  denominator  defined  on  page  157  is  the 
L.  C.  M.  of  all  the  denominators  of  the  fractions. 

4.  Find  the  least  common  denominator  and  then  add 

(1)  T2i>,  A,  A,  and  if;          (3)  i,  A,  A,  and  A; 

(2)  4,  A,  A,  and  A;        (4)  *»  T5s,  A.  and  A- 

§105.  Definitions  and  Principles. 

Thus  far  we  have  had  to  do  with  two  kinds  of  number: 
(1)  whole  numbers,  or  Integers,  and  (2)  fractional  numbers,  or 
Fractions. 

DEFINITIONS.— A  proper  fraction  is  a  fraction  whose  numerator  is  less 
than  its  denominator :  as  |,  £,  f . 

An  improper  fraction  is  a  fraction  whose  numerator  is  equal  to,  or 
greater  than,  its  denominator :  as  |,  f ,  I,  f. 

A  mixed  number  is  a  number,  such  as  2£,  12|,  7|,  that  is  composed 
of  an  integer  and  a  fraction. 

1.  Name  the  proper  and  the  improper  fractions,  the  mixed 
and  the  whole  numbers : 

4,  I,  A,  s>  -Y-,  f.  !.  11.  3A.  6-75,  A,  2- 

2.  Make  and  give  an  example  of  each  class. 

3.  How  many  times  is  1  contained  in  3?  in  18?  in  25?  in  any 
number? 


162  RATIONAL    GRAMMAR    SCHOOL    ARITHMETIC 

We  may  write  this  in  symbols,  thus: 


Any  whole  number  may  be  written  in  the  fractional  form  by 
writing  1  for  Us  denominator. 

4.  Write  the  following  numbers  in  fractional  form: 

12;     24;     32;     68;     «;     x 

5.  How  many  halves  are  there  in  1?  in  2?  in  5?  in  2£?  in  5J? 

6.  How  many  5ths  are  there  in  1?  in  2?  in  4?  in  6?  in  6f  ?  in 
7j? 

7.  Change  the  following  integers  into  6ths:  2;  5;  16;  29;  120. 

8.  Change  the  following  numbers  into  12ths:  2;  8;  12;  20; 
56;  128. 

0,  How  may  any  whole  number  be  changed  into  12ths?  into 
9ths?  into  24ths?  into  a  fraction  having  any  given  denominator? 

PRINCIPLE  V.  —  Any  integer  may  be  expressed  in  the  form  of 
a  fraction  having  a  given  denominator  by  multiplying  the  integer 
by  the  given  denominator  and  writing  the  product  over  that  denom- 
inator. 

10.  Express  6,  18,  17,  22,  39,  28  as  4ths;  as  7ths;  as  lOths;  as 
lOOths. 

11.  Change  into  6ths,  2;  4J;  24|;  37f. 

12.  Change  the  following  numbers   into   improper   fractions 
having  7  for  a  denominator: 

3|;     24f;     106f;     234?;     648. 

13.  How  can  you  change  any  mixed  number  into  an  improper 
fraction  whose  denominator  is  the  denominator  of  the  given  frac- 
tion? 

PRINCIPLE  VI.  —  A  mixed  number  may  be  expressed  as  an 
improper  fraction  by  multiplying  the  'whole  number  by  the  denom- 
inator, adding  the  numerator  to  the  product,  and  writing  the  sum 
over  the  given  denominator. 

14.  Eeduce  to  improper  fractions: 

2i;    6J;  12i;  18};  25f;  328|  ; 


COMMON    FRACTIONS 


163 


15.  Express  Y-  as  a  whole  number.     Change  V   to  a  mixed 
number. 

16.  Change  to  whole,  or  mixed,  numbers  the  following  improper 
fractions : 

f;  f;  i;  -¥•;  Y;  V;  V;  V;  V;  W;  If;  V;  V;  t- 

17.  How  may  any  improper  fraction  be  changed  to  a  whole, 
or  mixed,  number? 

PRINCIPLE  VII. — An  improper  fraction  may  be  changed  to  a 
whole,  or  mixed,  number  by  performing  the  indicated  division. 

18.  Reduce  to  whole,  or  mixed,  numbers  the  following: 

I;  V;  ff ;  7^;  V;  «*;  HI1- 

§106.  Addition  of  Fractions. 

1.  If  the  numbers  written  on  the  lines  of  the  drawing  are  the 
lengths  in  feet  of  the  inside  walls  of  the  house, 
how  many  feet  of  lumber  would  there  be  in  a 
baseboard  not  over  1"  thick,  1  ft.  wide  extend- 
ing entirely  around  the  inside  of  the  house, 
deducting  6  board  feet  for  doors? 

To  solve  this  problem  we  add  first  all  the 
whole  numbers  and  then  all  the  fractions. 
We  must  now  learn  how  to  add  such  common 
fractions  as  occur  here.  This  problem  will  be 
solved  later.  (See  problem  9  below.) 

2.  A  man  owns  f  of  an  acre  in  one  block  and  -J  of  an  acre 
in  another  block ;  how  much  land  does  he  own? 
This  problem  requires  us  to  add  f  and  -J. 

I.  GEOMETRICAL  SOLUTION.— Fig. 
90  (a)  is  divided  by  the  vertical  lines 
into  3  equal  parts  and  by  the  hori- 
zontal lines  into  8  equal  parts.  Into 
how  many  equal  parts  do  the  2  sets 
of  parallel  lines  divide  the  square 
(a)? 

Show  by  Fig.  90  (a)  that  |  =  \\. 
Show  by  Fig.  90  (6)  that  I  =  |i 


11* 


169/2 


FIGURE  89 


(a) 

i  =  H 

(6) 
i  =  11 

FlGUKE  90 


+  n  =  n  =  iif. 


Ans.  1J£  acres. 


164 


RATIONAL    GRAMMAR   SCHOOL   ARITHMETIC 


II.  ARITHMETICAL  SOLUTION.— What  is  the  least  common  denomi- 
nator of  }  and  5?  What,  then,  is  the  largest  fractional  unit  (therefore 
having  the  least  denominator)  in  which  both  g  and  %  can  be  exactly 
expressed? 

|  =  how  many  24ths?      I  =  how  many  24ths? 


3 

Ac 

7 

3- 
Id 

r  £ 

— 
}   2 

?T,  f  ,  and 

2 

i 

^ 

-  •  •  >(* 

ro 

.     Ans. 


acres. 


(a) 


M 


FIGURE  91 


I.  "GEOMETRICAL  SOLUTION.—  Show  from  Fig.  91  (a),  that  f  =  |f  ;  from 
Fig.  91  (5),  that   fr  =  &;    from  (c)  that  |  =  ff  ;    from  (d)  that  ft  =  f$. 

M  +  A  +  IS  +  II  =  «¥  =  if  i  =  HI-- 

II.  ARITHMETICAL  SOLUTION.—  The  least  common  denominator  of  the 
fractions  is  84. 

._       3X12,     2X4,2X28,     5x7      36  +  8+56  +  35 

f  "TlT-r  §  ~Ti2 


135 


7  X  12  '   21  X  4  '  3  X  28  '  12  X  7 


84 


The  last  step  consists  in  reducing  the  result  to  its  simplest  form. 

4.  T\  of  the  weight  of  a  specimen  of  soil  was  gravel,  and  % 
was  sand;  what  part  of  the  soil  by  weight  was  sand  and  gravel? 
Draw  a   figure   and   give  the  geometrical  solution.      Give   also 
the  arithmetical  solution. 

5.  One  side  of  a  triangle  is  2f  in.,  another  is  2f  in.,  and  the 
third  is  3|  in;  how  long  is  the  perimeter  of  (distance  around)  the 
triangle?     Add  first  the  whole  numbers,  then  the  fractions,  and 
finally  add  the  sums. 

6.  One  side  of  a  4-sided  figure  is  6f  in.  long,  a  second  side 
is  3f  in.,  the  third  7f  in.,  and  the  fourth  7T6^  in.     What  is  the 
perimeter  of  the  figure? 

7.  A  coal  dealer  bought  4  carloads  of  coal  of  the  following 
weights:  21£J  T.,  27J  T.,  29f  T.,  and  30f  T.     Find  the  com- 
bine i  weight,  in  tons. 


COMMON    FRACTIONS  165 

8.  Solve  these  problems  as  rapidly  as  you  can  work  accurately, 
using  your  pencil  merely  to  write  the  products  and  the  sums  : 

(1)  |  +  *  =?          (4)    4  +7S  =  ?          (7)  U  +  *  =  » 

(2)  |  +  1  =?          (5)  tt  +  ,V?          (8)   ^+T  =  ? 

(3)  T6T  +  T5*  =  ?          (»)*  +  «-?          (9)    T  +  T-» 

9.  Solve  problem  1  of  this  section. 

§107.  Subtraction  of  Fractions. 

1.  A  boy  had  $£  and  spent  &J;  what  part  of  a  dollar  did  he 
have  left? 

ARITHMETICAL  SOLUTION.  —The  least  common  denominator  is  2  X  5 

=  10. 


2.  William  is  4f  ft.  tall  and  James  is  4^  ft.  tall;  who  is  the 
taller  and  by  how  much? 

3.  A  man  owned  {£  A.  of  land  and  sold  4  A.  ;  how  much  land 
did  he  then  own? 

4.  From  £  Ib.  of  loam  f  Ib.  sand  was  removed  ;  how  much 
of  the  loam  remained? 

5.  A  man  having  I4J-  paid  a  debt  of  $2f;  how  much  money 
had  the  man  after  paying  the  debt? 

SOLUTION.—  J  =  240,  -5  =  $g.  As  &  is  less  than  Jg,  write  4^,  in  the 
form  3|J.  The  problem  is  then  8|$  —  2Jg  =  1,90  for  8  —  2  =  1  and  |J  — 
18  =  J9o- 

6.  Solve  these  problems: 

(1)  134   -  7|  -f  (4)  128f     -   97if  =  ? 

(2)  28f    -19|  =  ?  (5)   639f      -598T\  =  ? 

(3)  30,V-16J  =  ?  (6)  121716f-989i|  =  ? 

7.  A  tree  72f  ft.  high  is  broken  off  28T\  ft.  from  the  top  ;  how 
high  is  the  stump? 

8.  A  5-cent  piece  weighs  73  J-  gr.  and  a  quarter  dollar  weighs 
96^  gr.  5   how  much  more  does  a  quarter  weigh  than  a  5-cent 
piece? 

9.  The  dime  weighs  38T\  gr.     Before  1853  it  weighed  41  J  gr. 
By  how  much  was  the  weight  of  the  dime  reduced  in  1853? 


1G6  RATIONAL    GRAMMAR    SCHOOL    ARITHMETIC 

10.  A    silver    dollar   weighs   412-J    gr.    and    a    double   eagle 
weighs  516  gr.     What  is  the  difference  between  the  weight  of 
a  double  eagle  and  that  of  a  silver  dollar? 

11.  What  is  the  difference  between  the  weight  of  the  half 
dollar  (192^  gr.)  and  that  of  the  quarter  dollar  (96/7  gr-)? 

12.  Write  out  rapidly  the  values  of  these  sums  and  differences : 

(i)  i+*  =?       (0)  I  +  t  =?       (ii)  !  +1  =? 

(2)  i  -  A-  '  (?)    4     -  T\  =  ?  (12)   A-  A  -  ? 

(3)  t  -  A  -  '         (8)  T\  +  T*J  •  ?          (13)  A  +  A  =  ? 

(4)  t+V  =?        (9)  4  +  T=?          (14)  f+l-» 
(«)V-V-»       (10)=-    -T=?          US)  S  -7-' 

13.  Make  a  rule  for  finding  quickly  the  sum,  or  the  difference, 
of  any  two  fractions. 

NOTE. — This  latter  rule  may  be  formulated  thus, 

1st  num.  X  2d  den.  -\-  2d  num.  X  1st  den. 

1st  den.  X  2d  den. 
and 

1st  num.  X  3d  den.  —  2d  num.  X  1st  den. 

difference  =  —  . 

1st  den.  X  2d  den. 

The  result  must  always  be  reduced  to  the  simplest  form. 

14.  In   three   successive  runs,  a  train  loses   ^,  |,  and  %  hr. 
How  much  time  does  it  lose  in  all? 

15.  f  of  a  man's  salary  is  spent  for  clothing,  £  for  board  and 
lodging,  £  for  books  and  stationery,  T*f  for  traveling  expenses. 
What  part  of  his  salary  was  spent  for  these  purposes?  what  part 
remained? 

16.  In  the  six  days  of  one  week,  a  man  works  8-J-  hr.,  9f  hr., 
8f  hr.,  9£  hr.,  8f  hr.,  lOf  hr.      What  is  the  whole  number  of 
hours  of  work  for  the  week?    What  are  his  wages  at  $.30  an  hour? 

17.  Four  children  are  to  do  a  piece  of  work.    Three  of  them  do 
{,  £,  and  |  of  it.     What  part  is  left  for  the  fourth? 

18.  f ,  J,  and  T\  of  a  man's  money  are  invested  in  three  different 
enterprises.     What  part  of  his  money  is  invested?     What  part  is 
free? 

19.  A  merchant  sold  T"V  of  a  gross  of  buttons  to  one  customer, 
and  ££  of  a  gross  less  to  another.     What  part  of  a  gross  did  the 
second  customer  buy? 


COMMON    FRACTIONS  167 

20.  A  owns  T\  and  B  ^  of  an  estate.    How  much  more  of  the 
estate  does  A  own  than  B? 

21.  A  tank  of  oil  is  f  full.    If  £  of  the  contents  of  the  tank  is 
drawn  off  and  then  -fa  of  the  remainder,  what  part  is  left? 

22.  A  owns  f }  of  a  mill.     B's  interest  is  |f  of  the  mill  less 
than  A's.     What  part  of  the  whole  does  B  own? 

23.  At  $1  a  day  how  much  money  does  a  man  earn  by  working 
5f  da.,6f  da.,  and  8|  days? 

24.  A  boy  lives  where  school  is  taught  1100  hr.  a  year.    He  is 
compelled  by  sickness  and  other  causes  to  lose  from  month  to 
month  the  following  numbers  of  hours : 

115|;  39|;  15f;  6U;  4*5  3|;  5f;  18J;  89TV 
How  many  hours  did  he  lose  in  all?     What  part  of  the  whole 
school  year  did  he  lose? 

§108.  Multiplying  a  Fraction  by  a  Whole  Number. 

1.  6  times  3  mi.  =  how  many  miles. 

2.  8  x  4  yd.  =  ?  yards. 

3.  8  x  4  fifths  =  ?  fifths. 

4.  5  x  f  =  .- ;  what  is  the  value  of  x? 

5.  9  x  -fa  =  -^-;  what  is  the  value  of  #? 

6.  A  whole  is  divided  into  11  equal  parts,  and  5  of  them  are 
taken.     What  fraction  represents  the  part  of  the  whole  which  is 
taken? 

7.  What  fraction  would  represent  3  times  this  part? 

8..  Replace  the  letter  in  each  of  these  equations  by  its  correct 
value;  the  sign  (x)  should  be  read  "times"  in  these  problems. 

(l)6xf  =  f  <5)6xt>f  (9)2xT\  =  i 

(2)  4  x  1  -  f  (G)  3  x  |  =  f  (10)  6  x  Y-  T 

(3)  3  x  i  =  £  (7)  4  x  *  =  f  (11)  5  x  A  =  £ 

(4)  2  x  f  =  f  (8)  2  x  $  =  f  (12)  2  x  &  =  £ 

In  multiplying  these  fractions   by  the  whole   numbers  what 
change  was  made  in  the  numerator  to  get  the  product? 

Make  a  rule  for  multiplying  a  fraction  by  a  whole  number. 


168  RATIONAL   GRAMMAR   SCHOOL    ARITHMETIC 

9.  Apply  your  rule  to  these  problems,  reducing  to  whole  or 
mixed  numbers  all  the  products  that  are  improper  fractions : 

(1)  3  x  |  =  (5)  11  x  A  =  (9)     6  x  1  = 

(2)  5  x  f  =  (6)  9  x  TV  =  (10)     8  x  A  = 

(3)  7  x  |  =  (7)  4  x   I  =  (11)   10  x  f  = 

(4)  9  x  f  =  (8)  6x1-  (12)     9  x  «  = 

2      5       10 

NOTE.— In  such  as  (7)  use  cancellation,  thus:  £  x  ~-=  -5  =  3J. 

p         o 

3 

10.  Solve  these  problems  and  make  a   rule  for  multiplying 
a  fraction  quickly  by  its  denominator : 

(1)  6  x  |  =  ?  (5)  12  x  A  =  ?  (9)  75  x  4f  =  ? 

(2)  9  x  |  =  ?  (6)  18  x  if  =  ?  (10)  b   x  -JL  =  ? 

(3)  5  x  f  =  ?  (7)  21  x  if  =  ?  (11)  m  x  -£-  =  ? 

(4)  7x|  =  ?  (8)  48  x  W  =  ?  (12)  a   x  -£-  =  ? 

NOTE.— In  all  such  problems  use  cancellation,  thus:  /7  x  •=  =  5. 

11.  Give  the  values  of  the  letter  in  each  of  these  problems  and 
make  a  rule  for  multiplying  a  fraction  quickly  by  some  factor  of 

7  *7 

its  denominator;  (use  cancellation  thus :  %  x  -^  =  -^  =  l|). 

4 
(l)3x*=f;        (3)12x¥V  =  f;       (5)25xTW  =  f; 

(8)«*A  =  ?5        (4)     9xA  =  f'        (6)13xH=f 
State  the  rule. 

PRINCIPLE  VIII. — A  fraction  is  multiplied  by 

1.  Multiplying  its  numerator  ~by  the  multiplier;  or, 

2.  Dividing  its  denominator  by  the  multiplier. 

NOTE. — The  usefulness  of  the  method  of  cancellation  consists  in  mak- 
ing these  two  processes  undo  each  other.     (See  problem  11). 

QUERY. — When  will  the  second  method  (Principle  VIII,  2), 
be  the  more  convenient? 

A  fraction  is  multiplied  by  its  denominator  by  dropping  its 
denominator. 


OOMMOST    FRACTIONS  169 

§109.  Multiplying  a  Mixed  Number  by  a  Whole  Number. 

1.  What  will  bo  the  cost  of  12  yd.  of  muslin  at  8f  cents? 
SOLUTION.— If  1  yd.  costs  8jj  f  what  will  12  yd.  cost? 

12  X  8|  means  12  X  8  -f  12  X  §  =  96  -f  8  =  104.     Ans.  §1.04. 

NOTE. — It  is  important  to  bear  in  mind  in  dealing  imth  such  expressions 
as  12  X  8 -f  12  X  |,  which  contain  both  signs  (X)  and  (-f),  that  the 
indicated  multiplications  must  be  performed  first,  and  then  the  additions. 
Thin  fact  is  usually  expressed  by  saying,  "the  multiplication  sign  (X) 
takes  precedence  of 'the  addition  sign  (+)." 

2.  13  x  4|  -  ? 

4 

13 

JT"  __  1Q  v  A      Multiply  tlie  whole  numbers  in  the  usual  way. 
71  =  18  x  I      13  X  8  =  ¥  =  7|,  as  in  problem  H,  §108. 

59|  =  13  X  4* 

PROBLEMS 

Solve  the  following,  using  the  more  convenient  method  in  each 
case,  employing  cancellation  whenever  it  shortens  the  work: 

1.  5xii  =  5.   9x  JJ  =  9.   2x-i/  = 

2.  3  x  |   =  6.    9  x  f   =  10.   3  x  GJ  = 

3.  5  x4|  =  7.   9  xTV  =  11.  9x^8-  = 

4.  2x2J  =  8.   9x  -i  .  12.   8  x  T\  = 

13.  Find  the  cost  of  25  yd.  of  cloth  at  $f  per  yard. 

14.  A  earns  II :{,  and   B  earns  4  times  as  much.     B  earns  $#. 
Find  x. 

15.  A  grocer  sold  J  bu.  of  apples  to  one  customer,  and  5  times 
that  quantity  to  another.     He  sold  y  bu.  to  the  second  customer. 
Find?/.     If  apples  were  $.40  a  peck  what  amount  of  money  did 
the  grocer  receive  from  the  first  customer?  from  the  second? 

16.  If  a  man  can  cut  £  of  a  cord  of  wood  in  one  day,  how 
much  can  he  cut  in  6  da.,  working  at  the  same  rate? 

17.  One  boy  lives  T5g-  mi.  from  school  and  another  4  times  as 
far.     What  is  the  distance  from  school  to  the  second  boy's  home? 

18.  Solve  problems  based  upon  the  following  facts : 
An  iron  tube  weighs  f  Ib.  per  foot. 

f  of  the  weight  of  water  is  oxygen. 

The  circumference  of  a  bicycle  wheel  is  6J-J-  feet. 

A  west  wind  moved  47f  mi.  per  hour. 


170  KATIONAL    GRAMMAR    SCHOOL    ARITHMETIC 

§110.  Multiplying  a  Whole  Number  by  a  Fraction. 

The  whole  number  is  here  the  multiplicand. 

DEFINITION.  —  To  multiply  a  whole  number  by  a  fraction  means  to 
divide  the  multiplicand  into  as  many  equal  parts  as  there  are  units  in 
the  denominator,  and  to  take  as  many  of  these  equal  parts  as  there  are 
units  in  the  numerator  of  the  multiplier. 

ILLUSTRATION.  —  12  multiplied  by  |  means  that  5  of  the  6  equal  parts 
of  12  are  wanted.  -J  times  12=  2,  and  |  times  12  =  5  X  2  =  10. 

|  times  12  and  |  of  12  mean  the  same  thing. 


" 


1.  Solve  these  problems,  reading  the  sign  (x)  "times 

(1)  f  x  10  =  ?  (4)    f  x  15  =  ?  (7)    f  x  45  =  ? 

(2)  f  x  25  =  ?  (5)   f  x  28  =  ?  (8)  T\  x  26  =  ? 

(3)  1  x  32  =  ?  (6)  r6T  x  33  =  ?  (9)  T\  x  47  =  ? 
SUGGESTION.—  In  (9),  j>r  of  47  =  4^I(and  7  x4ft  =  28?}  = 


2.  Solve  the  following  : 

(1)  T\  x  50  =  ?  (3)  tJ  x  105=  ?  (5)  ^  x  110  =  ? 

(2)  fj  x  39  =  ?  (4)   ff  x  220  =  ?  (6)  6|  x    28  =  ? 
SUGGESTION.—  6£  x  28  means  6  X  28  -f  I  X  28  =  168  -f  21g  =  189£. 
NOTE.  —  See  note  to  problem  1,  page  169. 

§111.  Factors  May  Be  Interchanged. 

In  Fig^  92  (a)  the  rectangle  is  supposed  to  be  ^  in.  wide  and 
5  in.  high.     The  area  is  then  5  times  -J  sq.  in.  =  2£  square  inches. 

In  (b)  the  rectan- 

5     _  .  -  .      gle  is  5  in.  long  and 
I          I  i  in.  high.     Its  area 

is  %  times  5  sq.  in.,  or 
^  of  5  sq.  in.  =  2^  sq. 
in-.,  as  is   plain   from 
a  Fig.  92. 

This  exemplifies 
the  truth  that  when 
a  fraction  and  a  whole 
number  are  to  be  mul- 
tiplied it  makes  no 
difference  in  the  nu- 
FIGURB  92  merical  value  of  the 

product  which  of  the  factors  we  regard  as  the  multiplicand. 

The  name  of  the  result  is  determined  by  the  conditions  of  the 
problem. 


COMMON    FRACTIONS  171 

Letting  f-  (read  "x  divided  by  i/")  stand  for  any  fraction,  and  a 
stand  for  any  whole  number,  state  the  following  principle  in. 

symbols : 

The  product  of  a  whole  number  by  a  fraction,  or  of  a  fraction 
by  a  whole  number,  equals  the  product  of  the  whole  number  by 
the  numerator,  divided  by  the  denominator. 

PROBLEMS 

1.  Find  the  cost  of  2J  yd.  of  cloth  at  $3. 

2.  A  man  had  $75;  he  spent  f  of  it  for  a  bicycle,  and  £  of 
the   remainder   for  clothing.     Find  the  amount  spent,   and  the 
amount  he  had  left. 

3.  A  gallon  contains  231  cu.  in.     How  many  cubic  inches  in 
f  gal.?  in  |  gallon? 

4.  A   cubic    foot    of    granite    weighs    170   Ib.       How    many 
pounds  in  of  cubic  feet? 

5.  T\  of  a  plot  of  ground  containing  918  sq.  rd.  was  fenced  for 
a  garden.     The  garden  contained  how  many  square  rods? 

6.  Make  and    solve  problems,    based  upon    items   personally 
obtained,  or  upon  those  given  below: 

An  avoirdupois  pound  of  gold  is  worth  about  $348f . 

A  bale  of  cotton  ordinarily  weighs  450  Ib.  Price  17|^  per 
pound. 

A  tank  contains  12G  gal.  linseed  oil.     Price  $.62  per  gallon. 

Corn  is  quoted  and  sold  @  50J^  a  bu.  Wheat  is  quoted  and 
sold  @  77f  ^  a  bushel.  Oats  are  quoted  and  sold  @  32f  <fi  a  bushel. 

To  plow  an  acre  with  a  plow  cutting  a  furrow  10"  wide  (a  10" 
plow)  a  horse  must  walk  9T\  mi. ;  with  a  12"-plow,  a  horse  must 
walk  8J  mi. ;  with  a  15"-plow,  6f  miles. 

For  a  team  drawing  a  plow,  a  distance  of  16  mi.  is  a  fair  day's 
work  and  a  distance  of  18  mi.  is  a  large  day's  work.  The  follow- 
ing table  shows  to  what  fractional  part  of  an  acre  a  furrow  1  mi. 
long  and  of  the  indicated  widths  is  equivalent : 


WIDTH  OF  FURROW 

EQUIVALENT 

WIDTH  OF  FURROW   EQUIVAL 

12" 

A  A. 

16" 

W  A. 

13" 

iV3e  A. 

17" 

rV*  A. 

14" 

-            A 

nV  A. 

18" 

A  A. 

15" 

ft  A. 

20" 

IM 

172 


RATIONAL    GRAMMAR    SCHOOL    ARITHMETIC 


at 


a 


L— G 


G- 


4/5 
FIGURE  93 


B- 


;112.  Multiplying  a  Fraction  by  a  Fraction. 

1.  What   is   the  area  of   a  rectangle 
3"x5",  Fig.  93  (1)?  6"x8"?  7'x  8'?  2  rd. 
x  80  rd.?  16x20?     a  x  J? 

2.  How  many  square  feet  in  the  area 
of  a  rectangle  -J-'x  8'?     f'x  12'?     f'x  16'? 
TYx  30'?     (Use  cancellation. ) 

3.  What  is   the   area  of   a  rectangle 
f"x-f"?     [See  Fig.  93  (2).] 

SOLUTION.—  ABCD  is  a  rectangle  i"  long 
and  f"  wide. 

AEGH  is  a  square  inch. 
What  part  of  a  square  inch  is  ABKH1 
What  part  of  ABKH  is  ABCD  ?     Point  out 
on  the  figure  the  part  that  represents  §  of  £  of 
1  square  inch? 

Into  how  many  small  rectangles,  such  as 
a,  do  the  dotted  lines  divide  the  square  inch? 
What  part  of  the  square  inch  is  one  of  the  small  rectangles,  a? 
How  many  of  these  are  there  in  the  given  rectangle  ABCD  ? 
The  area  of  ABCD  is  then  what  part  of  a  square  inch? 

We  may  write  the  two  expressions  for  the  area  equal,  thus : 
I  X  *  =  ft- 

We  see  then  that,  if  our  law  for  finding  the  area  of  the  rectangle, 
which  we  have  found  to  hold  for  integers,  is  to  hold  for  fractions  also, 
we  must  define  both  |  X  |  and  £  X  I  to  be  the  same  as  |  of  ^.  Each  is 
equal  to  &. 

It  is  clear  that,  whatever  the  fractions  to  be  multiplied  may 
be,  the  square  unit  [sq.  in.  in  Fig.  93  (2)]  may  be  divided  by  the 
cross  lines  into  as  many  equal  parts  as  there  are  units  in  the  prod- 
uct of  the  denominators,  and  that  there  will  be  as  many  of  these 
equal  parts  in  the  given  rectangle  as  there  are  units  in  the  prod- 
uct of  the  numerators. 

Hence  our  fractions  may  be  multiplied  by  merely  writing  the 
product  of  the  numerators  over  the  product  of  the  denominators 
with  the  dividing  line  between.  Cancellation  should  be  used 
whenever  possible. 


4.  Draw  a  square  inch  and  find  these  products  as  above: 

.  (i)  r  x  r  =  ?     w  f  "  x  t"  -  ?     (7)  r  x  j  "  =  ? 

(2)  r  x  i"  =  ?     (5)  j"  x  r  =  ?     (8)  r  x  f  =  ? 

(3)  r  x  f  =  ?     (6)  f 


COMMON    FRACTIONS  17o 

5.  By  drawing  and  dividing  a  square  inch,  as  in  Fig.  93,  show 
that  the  following  equations  are  true : 

(1)  f  x  T<V  =  f  of  A;     (3)   f  x  A  =  A  of  f ;     (5)   f  x  A  =  •?!> 
(2)i5ix  f  -A  <>f  -|;     (4)   T5Txf  =  -|of  T\;     (G)   A  *  !  =  H 

To  find  -\  of  A,  we  take  a  fractional  unit  which  is  %  of  A 
(the  fractional  unit  of  A)>  an^  use  the  same  number  (5)  of  them, 
thus  obtaining  A-  But  f  of  A  =  6  x  |  of  A  =  G  x  A  »  f  f  As 
$  x  T5r  =  f  °^  A  >  f  x  i T  =  f  T-  Similar  reasoning  would  show  also 
that  A  x  i  =  TT-  Now  we  recall  that  the  numerator  (30)  of  the 
product  (AX4)  was  obtained  by  multiplying  the  numerators 
(6  and  5)  of  the  factors  (T5T  and  ^).  How  was  the  denominator 
(77)  of  the  product  obtained? 

G.  State  a  rule  for  quickly  multiplying  a  fraction  by  a  fraction. 

7.  Solve*  these  problems  as  rapidly  as  you  can  work  accurately: 
.        (1)  fxi         (5)  f  x|         (9)  £xf         (13)   fxf 

(2)  ixf    (6)  ixf   (10)  fxf    (14)  }x| 

(3)  fxf    (7)  f  x  f   (11)  f  x  |    (15)  f  x  f 

(4)  |x|    (8)  fxf   (12)  fxf    (1G)  |x| 

8.  Letting  •£    and  -J-  denote  any  two  fractions,  state  the  fol- 
lowing principle  in  symbols : 

The  product  of  any  two  fractions  is  a  fraction  whose  numer- 
ator is  the  product  of  the  numerators  and  whose  denominator  is 
the  product  of  the  denominators. 

§113.  Multiplying  a  Mixed  Number  by  a  Mixed  Number. 

1.   How  many  square  feet  in  the  area 
of  a  rectangle  4f "  x  o-|-"? 

(1)  How  long  is  a  (Fig.  94  )?  how  wide? 
What  is  its  area? 

(2)  Answer  similar   questions   for    b, 
c,  and  d. 


(3)  How  long  is  the  entire  rectangle?  5  ' 

how  wide?     What  is  its  area?  FIGURE  94 

Since  the  area  of  the  entire  rectangle  equals  the  sum  of  the  areas  of 
its  parts  we  may  write  (remember  X  takes  precedence  of  +;  see  note, 
problem  1,  page  169): 

4f  x5i»=4xr>+»  X5  +  4X  £ +  5  X  5  =  20+ 8f +  2 +  f  =  26J. 

Ans.  26|  sq.  ft. 

*  Cancel  whenever  it  Is  possible. 


174  RATIONAL    GRAMMAR   SCHOOL   ARITHMETIC 

Point  out  on  figure  the  areas  that  represent  all  the  parts  of  the 
product. 

2.   Find  the  cost  of  9£  T.  hard  coal  at  $7f  . 

FIRST  SOLUTION.—  If  1  T.  costs  $7f  ,  9|  T.  will  cost  9^  x  $7f  . 

CONVENIENT  FORM  EXPLANATION 

7|  9J  X  7f  means-  9  X  7|  -f  fc  X  7f. 

9£  |  X  7|  means  £  of  7|. 

63   =9x7  9X7| 

6   =  9    X       =s^  7 


=    \  r  X    !  9£  X  7|  = 


73| 

SECOND  SOLUTION.—  9J  T.  =  -V*-  T.  ;  $7f  =  $V 

J£  X*£  =  fi|a,  or  731.     Ans.  $73|. 


Find  areas  of  surfaces  having  the  following  dimensions  : 

3.  44f  ft.  x  28f  feet. 

4.  36f  ft.  x  27f  feet. 

5.  24f  ft.  x  18|  feet. 

6.  45f  ft.  x  30£  feet. 

7.  27i  ft.  x  18f  feet. 


Find  the  cost  of  the  following  items  : 

8.  24f  doz.  eggs  @  $.16|. 

9.  46|  gal.  of  oil  @  $.12|. 

10.  15^  yd.  of  cloth  @  $.66|. 

11.  52}  Ib.  of  sugar  @  $.05|. 

12.  265f  M.  of  pine  flooring  @  $35J. 

Make  original  problems  from  the  following  items  : 

A  cubic  inch  of  water  weighs  252||  grains. 

A  water  tower  is  2f  J  ft.  higher  than  the  mound  upon  which 
it  is  built.  The  mound  is  81-J-  ft.  high. 

69-J  statute  miles  -  I  degree  of  longitude  at  the  equator. 

A  pine  tree  87  ft.  high  has  no  branches  for  £f  of  its 
height. 

-|i  of  a  library  containing  55,447  volumes  was  destroyed  by 
fire. 


COMMON    FRACTIONS  175 

Flaxseed  cost  $lf  per  pound  when  a  certain  linseed  oil  factory 
bought  supplies. 

A  spring  furnishes  28T\  bbl.  of  water  daily. 

A  certain  vessel  sails  llf  mi.  per  hour,  on  an  average. 

A  room  is  32  ft.  long  and  24 f  ft.  wide.  Painting  costs  $-5^  per 
square  foot. 

A  cubic  foot  of  water  weighs  62.5  pounds. 

Gold  is  19^  times  as  heavy  as  water. 

30J  sq.  yd.  =  1  square  rod. 

§114.  Dividing  a  Fraction  by  a  Whole  Number. 

OEAL   WORK 

1.  What  is  4  of  40A.?     40A.  divided  by  8  equals  what? 

2.  \  of  35  Ib.  =  ?     35  Ib.  +  7  =  ? 

3.  TV  of  80  =  ?     80  -»•  10  =  ? 

4.  i  of  3  fourths  =  ?     3  fourths  -*-  3  =  ? 

5.  |of|A.  =  ?     I-A.  +5=? 

6.  Jof  f  ft.  =  ?     ^  ft.  +3=  ? 

7.  £  of  f  J  in.  =  ?     f  J  in.  +  9  =  ? 

8.  Compare  these  fractional  units,  or  unit  fractions: 

(1)  i  is  what  part  of  J?  (7)  ^  is  what  part  of  £? 

(2)  I  is  what  part  of  £?  (8)  TV  is  what  part  of  £? 

(3)  I  is  what  part  of  i?  (9)  J  is  what  part  of  £? 

(4)  TV  is  what  part  of  1?          (10)  TV  is  what  part  of  £? 

(5)  |  is  what  part  of  4?          (11)  TV  is  what  part  of  i? 

(6)  i  is  what  part  of  J?          (12)  ^  is  what  part  of  J  ? 

9.  How  is  the  size  of  a  unit  fraction  changed  by  multiplying 
its  denominator  by  2?   by  3?   by  4?  by  10?  by  100?  by  a?  by  m? 

10.  By  what  must  you  multiply  the  first  fraction  in  each  of 
these  pairs  to  get  the  second? 

(1)  §  and  |?        (4)  A  and  f  ?         (7)  /,V  <«id  !i? 

(2)  T\and  |?         (5)  «  and  it?         (8)  7^  and  f? 

(3)  A  and  f?         (6)  ff  and  f  f  ?          (9)  ^  and  T-f? 

11.  Make  a  rule  for  dividing  a  fraction  by  3  without  changing 
its  numerator ;  by  5 ;  by  28 ;  by  a. 


176  RATIONAL    GRAMMAR    SCHOOL    ARITHMETIC 

WRITTEN    WORK 

1.  What  number  should  stand  in  place  of  x  in  these  examples? 

(1)  if-.    3  =  /T;     (4)   V/+    16  =  A;     (7)   lf-9  =  T\; 

(2)  -V  -15=  f  ;     (5)-W-    25  =  ft;     (8)    T+«=T; 


2.  Solve  these  problems  by  finding  what  number  should  stand 
in  place  of  x  : 

(1)  1-3  =  4;         (3)  tf  +   8-fA;       (5)  f^-    3-i±i; 
(2)2*4  =  1;        (4)V+a5  =  -£;       (6)  tW  +  «  -41- 

3.  In  what  two  wi*ys  may  a  fraction  be  divided  by  a  whole 
number?     When  would  you  use  the  method  of   problem  1?   of 
problem  2? 

PRINCIPLE  IX.  —  A  fraction  may  be  divided  by  a  whole  num- 
ber, (I)  by  dividing  its  numerator  by  the  whole  number,  without 
changing  its  denominator,  or  (2)  by  multiplying  its  denominator 
by  the  whole  number,  without  changing  its  numerator. 

NOTE.—  The  note  after  Principle  VIII,  page  168,  applies  here  also. 

PROBLEMS 

1.  Divide  f  Ib.  of  shot  equally  among  5  boys,   and  give  the 
weight  of  one  share. 

2.  -J  Ib.  of  maple  sugar  is  to  be  equally  divided  among  3  chil- 
dren.    What  part  of  a  pound  will  each  receive? 

3.  f  of  a  piece  of  work  can  be  done  in  8  da.     What  part  of 
it  can  be  done  in  1  day? 

4.  Make  and  solve  problems  based  upon  the  following  items, 
showing  work  in  each  case  : 

$f|-  to  buy  lace  costing  $5  a  yard. 

$J-J-  to  buy  apples  costing  $3  per  barrel. 

f  of  an  estate  divided  among  4  heirs. 

5.  If  |  Ib.  of  coffee  is  divided  into  4  equal  portions,  how  much 
is  there  for  each  portion? 

6.  3  Ib.  of  butter  cost  If;  1  Ib.  cost  how  much? 


COMMON   FRACTIONS  177 

7.  A   strip   of  tin-foil   -f  ft.   long   is   to  be   cut  across   into 
4  equal  pieces.     What  is  the  length  of  one  strip? 

8.  4  children  have  a  garden  containing  f  A.     If  the  children 
receive    equal     portions,     what     part    of     an    acre    does    each 
receive? 

9.  4  men  own  f  of  an  estate,  equally.     What  part  belongs 
to  each  man?     If   the  estate   is  valued  at  $36,000,  how  much 
money  is  represented  by  each  part? 

10.  If  -1!5  A.  is  divided  equally  among  7  men,  how  many  acres 
are  there  for  each  man? 

§115.  Dividing  a  Mixed  Number  by  a  Whole  Number. 

1.  7  strips  of  carpeting  cover  a  room  5£  yd.  wide.     What  is 
the  width  of  one  strip? 


5J  yd.  = 

\*  yd.  -7  =  f  yd.,  the  width  of  one  strip. 

5J  is  what  kind  of  number? 

Before  dividing  what  change  was  made?  Was  the  division 
performed  by  multiplying  the  denominator  or  by  dividing  the 
numerator? 

2.  A  boy  earned  $4^  in  6  da.  What  did  his  earnings  average 
per  day? 


2 

How  was  the  dividing  done  in  this  case? 

3.  Make  a  rule  for  dividing  a  mixed  by  a  whole  number. 

To  divide  a  mixed  number  by  a  whole  number  first  change  the 
mixed  number  to  an  improper  fraction  and  then  proceed  as  in  the 
division  of  a  fraction  by  a  whole  number. 

PROBLEMS 

1.  36f  yd.  of  cloth  were  made  up  into  6  ladies'  suits.  If  the 
suits  contain  the  sarno  number  of  yards,  how  many  yards  are 
there  in  each  suit? 


178  RATIONAL    GRAMMAR    SCHOOL    ARITHMETIC 

2.  A  ceiling  16   ft.    wide    contains   233£   sq.   ft.      Find   the 
length. 

233£  ft.  =  if  *  feet. 
175 

mtfr 

™~  *  16  =  3^1ff =  W  =  14iV    14iV  =  14'  7"  which  is  the  length 

4 
of  the  room. 

3.  Solve  the  following  problems : 

(1)  14f  bu.  -*-    8  =  (6)  $12f  -i-  5  = 

(2)  9TV  yd.  +    6  =  (7)  45ft  +  5  = 

(3)  16f  hr.  +  12  =  (8)  ^  +  4  = 

(4)  44ft  lb.+    9  =  (9)  14fr  -«-  7  = 

(5)  39f  mi.  *    7  =  (10)  66|    ^  8  = 

NOTE. — In  problems  like  (7)  and  (9),  in  which  the  integral  part  of  the 
dividend  exactly  contains  the  divisor,  divide  the  whole  number  and  the 
fraction  separately  and  add  the  quotients. 

4.  Find  the  lengths  of  the  surfaces  whose  areas  and  widths  are 
given  here : 

AREAS  WIDTHS  LENGTHS 

(1)  212J  sq.  ft.  17  ft. 

(2)  261±      "  14  " 

(3)  406J      "  25   " 

(4)  164J      "  13  " 
Make  problems  based  on  these  items : 

5.  A  ship  sails  246T3¥  mi.  in  36  days. 

6.  A  man  paid  $195£  for  72  clays'  board. 

7.  A  field  contains  2172^  sq.  rd.     One  side  is  40}  rd.  long. 

8.  15  Ib.  of  fresh  salmon  cost  $2.71f . 

§116.  Dividing  Any  Number  by  a  Fraction. 

(A)  Dividend  a  whole  number. 

1.  How  many  fib.  packages  can  be  filled  from  3  Ib.  tea? 

(1)  How  many  i  Ib.  packages  can  be  made  from  1  Ib.?     How 
many  f  Ib.  packages  can  be  made  from  1  pound? 

(2)  How  many  f  Ib.  packages  can  then  be  made  from  3  Ib. 
tea? 


COMMON    FRACTIONS  179 

SOLUTION.  —  \  is  contained  in  1,  4  times,  f  is  contained  in  1,  £  this 
number  of  times,  f  is  then  contained  in  1,  |  times.  How  many  times 
is  it  contained  in  3? 

i  -*-!  =  !;  4x3  =  4. 

p 
Or,  since  f  X  §  =  1,  f  is  contained  in  1,  |  times. 

.2.  A  boy  earns  at  the  average  rate  of  $f  per  day  selling  papers. 
How  many  days  did  it  take  him  to  earn  $12? 

(1)  What  effect  does  it  have  on  a  fraction  to   multiply  its 
numerator?  to  multiply  its  denominator? 

(2)  What  effect  does  it  have  on  a  fraction  to  multiply  both  its 
numerator  and  its  denominator  by  the  same  number? 

(B)  Dividend  a  fraction. 

3.   Divide  f  by  f  . 
SOLUTION.— 

4x7  4x7 

4         ^  v  7  5  V  7       2ft 

-  Y^-J  ;  why?   [See  prob.  2  (2)  above.  ]     -2L.  =  -^  ;  why? 


T><~5  7X5 

(C)  Dividend  and  divisor  both  mixed  numbers. 

4.  I  paid  $49|  for  coal  @  $7f  per  ton.     How  many  tons  did  I 
buy? 

SOLUTION.— 

49|=^;      71=^;        49|  +  7f  = 

8 

248          ?£X)X4 

T 


_ 

31        ~  ^  -     6     -        T 

*  ^-X  5  X  * 

Ans.  6|  T. 

Let  us  now  seek  a  general  method  of  dividing  any  number  by  a 
fraction. 

5.  How  many  times  is  each  of  these  fractions  contained  in  1? 

i  ;  i  ;  i  ;  T  5  i  >  TO  ;  -rs  ;  A  ;  ^5  ;  70  ;  T^  o  ;  -^  j  v- 


180  RATIONAL   GRAMMAR   SCHOOL   ARITHMETIC 

6.  How  many  times  is  each  of  these  fractions  contained  in  1? 

I;  i;  4;  f  ;  I;  A;  A;  if;  H;  H;  T3zrV;  1- 

7.  "What  number  should  stand  in  place  of  the  letter  x  in  each 
of  these  products? 

(1)  |x|-z;         (4)    |x  f=z;         (7)  A><V=*; 

(2)  fxf  =  z;         (5)  AX¥  =  *;         (8)   T  x  4  =  *; 

(3)  *xf-a;         (6)HxH-*;         (9)    7x4=*. 

8.  What  number  should  stand  in  place  of  the  letter  y  in  these 
products? 

(l)|xy  =  l;          (4)Axy  =  l;          (7)  ffxy  =  l; 

(2)  fxy-1;          (5)  Ifxy-lj          (8)    -Jxy-1; 

(3)  fxy-1;          (6)    fx  */  =  !;          (9)    ixy-1. 


DEFINITION.—  1  divided  by  f  equals  f,  and  IV  §  =  1;  l-*-8  = 
1  -*-  1  =  8  and  so  with  other  numbers.     Dividing  1  by  any  number,  whole 
or  fractional,  is  colled  inverting  the  number. 

The  reciprocal  of  any  number  is  the  number  inverted. 

9.  To  what   number  is  the  product  of  any  number  and  its 
reciprocal  equal? 

10.  Examine   your  answers   to  problem  7  and  state  how  to 
find  quickly  the  number  of  times  any  fraction  is  contained  in  1. 

11.  If  a  fraction  is  contained  in  1  f  times,  how  many  times  is 
it  contained  in  2?  in  6?   in  12?   in  £?  in  f  ?  in  J?  in  any  number? 
in  a? 

NOTE.—  7  may  be  written  f,  12  may  be  written  *j*,  and  so  on.  Con- 
sequently 7  and  f  are  reciprocals,  as  are  also  12  and  fa;  25  and  0*5,  and 
so  on. 

12.  What   is   actually    done    by   inverting  the   divisor  in   a 
problem  in  division  of  fractions?     (See  Definition  above.) 

13.  If,  then,  the  reciprocal  of  a  fraction  is  multiplied  by  6, 
what  operation  is  performed? 

14.  Can  you  make  a  rule  for  quickly  dividing  any  whole  num- 
ber by  a  fraction? 

PRINCIPLE  X.  —  The  reciprocal  of  any  fraction  shows  how 
many  times  the  fraction  is  contained  in  1. 

PRINCIPLE  XI.  —  Any  number,  mixed,  integral,  or  fractional, 
may  be  divided  by  a  fraction  by  multiplying  the  number  by  the 
reciprocal  of  the  divisor. 


COMMON    FRACTIONS  181 

15.  Why  do  we  invert  the  divisor?     Why  do  we  multiply  the 
reciprocal  of  the  divisor  by  the  dividend? 


PROBLEMS 


1.  f  yd.  of  silk  costs  $f  .     Find  the  cost  of  1  yard. 

2.  At  $1£  a  yd.  how  much  cloth  can  be  bought  for  If? 

3.  At  $6^  per  hundred  how  much  beef  can  be  bought  for 


4.  How  many  meals  will  Q  bu.  of  oats  supply,  if  T3T  bu.  are 
fed  at  a  meal? 

s.,f  5.  How  many   pieces   of   wire 

*    .          -  *        *]  I  ft.  long  can  be  cut  from  a  piece 

!    .    !    !  ^  -  '  -  49*  ft.  long? 

#•  6.  Measure  If  ft.  by  f  ft. 

FIGURE  95  (See  Fig.  95.) 

Draw  lines  and  illustrate  the  following  : 

7.  2£  ft.  +  |  ft.  =  10.  3f  ft.  +  |  ft.  = 

8.  4f  ft.  +  A  ft-  =  11-  lift.*   ift.  = 

9.  2J  ft.  +  f  ft.  =  12.  Divide  3*  yd.  by  £  yard. 

13.  Divide  16J  yd.  by  2*  yd.  ;  by  If  yd.  ;  by  4*  yd.  ;  by  1J  yd.  ; 
by  |  yard. 

14.  Divide  *6|  by  *ft;  by  $^;  by  $f  ;  by  $f. 

15.  Find  the  number  of  strips  of  carpeting  running  the  long 
way  of  the  room  : 

WIDTH  OF  ROOMS  CARPETING 

22*  ft.  2*  ft.  wide. 

1  yd.  f  yd.  « 

18f  ft.  2*  ft.    " 

8i  yd.  |  yd.  « 

15  *  ft.  2*  ft.   « 

Make  and  solve  problems  based  upon  the  following  items  : 

16.  Portland  cement  $2f  per  bbl.  ;  $9  purchasing  fund. 

17.  Sidewalk   having  an    area    of    607£  sq.  ft.;    flagstones 
5£  ft.  square. 

18.  15|  gal.  maple  syrup;  f  gal.  measures. 

19.  29f  doz.  oranges;  -fy  doz.  oranges 


182  RATIONAL    GRAMMAR   SCHOOL    ARITHMETIC 

20.  The  ratio  of  two  numbers  is  f  ;  one  of  the  numbers  is  f. 

21.  A  man  bought  a  house  for  $1860.     The  house  cost  him  f 
as  much  as  he  paid  for  a  mill.     What  was  the  cost  of  the  mill? 

22.  My  library  contains  1075  volumes,  which  are  f  as  many 
as  a  friend's  contains.     How  many  volumes  are  in   my  friend's 
library? 

23.  Of  a  number  of  cars  of  grain  inspected  on  a  certain  day 
36  were  rejected.     This  number  was  Jf   of   the  entire  number 
inspected.     Find  the  number. 

fl  /7* 

24.  Let   -T-  and  —  denote  two  fractions,  and  state  the  follow- 

b  y 

ing  principle  in  symbols:  The  quotient  of  one  fraction  by 
another  is  equal  to  the  product  of  the  dividend  by  the  inverted 
divisor.  (Compare  Principle  XL) 

§117.  Complex  Fractions. 

Any  problem  in  division  of  fractions  may  be  written  in  frac- 
tional form,  the  dividend  being  written  above  and  the  divisor 
below  the  line. 

EXAMPLE.  —  The  problem,  to  divide  f  by  TBf,  may  be  written: 


Or,  the  problem,  to  divide  18|  by  6|,  may  be  written: 


DEFINITION.  —  Such  fractions,  containing  fractions  in  one  or  both  terms, 
are  called  complex  fractions. 

Principle  XI,  §116,  will  enable  us  to  solve  all  such  problems. 
If  either  the  divisor  or  the  dividend,  or  both,  are  mixed  numbers, 
they  should  first  be  reduced  to  improper  fractions. 

Following  this  principle  : 

£  =  t  x  Jf  =  H. 

The  second  step  becomes  unnecessary  if  we  notice  that  the 
product  of  the  outside  terms  (6  and  11),  called  the  extremes^  gives 
the  numerator,  and  the  product  of  the  inside  terms  (7  and  5), 

called  the  means,  gives  the  denominator  of  the  result. 


COMMON     FRACTIONS  183 

A  complex  fraction  is  simplified  when  its  terms  are  freed  of 
fractions,  and  the  resulting  fraction  is  expressed  in  its  simplest 
form. 

In  many  cases  factors  may  be  cancelled  from  both  numerator 
and  denominator  before  multiplying  the  fractions.  Cancel  when- 
ever possible. 

Show  that  the  method  explained  is  the  same  as  dividing  the 
product  of  the  extremes  by  the  product  of  the  means  in  the  given 
complex  fractions. 

PROBLEMS 

Simplify  the  following  complex  fractions,  cancelling  when  pos- 
sible : 

1. 
2. 
3. 

§118.  Joint  Effects  of  Forces. 

1.  A  force  of  23  Ib.  is  pulling  the  car  (Fig.  96)   toward  the 
right   and   another    of    12    Ib.    is    pulling   it 
toward  the  left.     If  the  car  is  free  to  move,  in 
FIGURE  96  which   direction  will  it  move?     What    single 

force  would  move  the  car  in  the  same  manner 
as  do  both  of  these  forces  pulling  at  the  same  time? 

2.  What  single   force   will  produce   the 
same  motion  of  the  car  (Fig.  97)  as  8x  Ib. 
acting  toward  the  right  and  5x  Ib.  acting 
at  the  same  time  toward  the  left? 

3.  What  single  force  will  move  the  car 
(Fig.   98)    in  the  same   manner   as   6m  Ib. 

FIGURE  98  pulling  against  2m  Ib.  will  move  it? 

When  a  force  of  18  Ib.  is  supposed  to  be  pulling  a  car  toward 
the  right)  it  will  be  written  thus:  R  18  Ib.  When  the  same  force 
pulls  toward  the  left)  it  will  be  written  thus:  L  18  Ib.  When 
two  forces,  R  16  Ib.  and  L  12  Ib.,  act  at  the  same  time,  their  joint 
effect  is  the  same  as  the  effect  of  the  single  force,  R  4  Ib. 


184  RATIONAL    GRAMMAR    SCHOOL    ARITHMETIC 

4.  What  would  bo  the  joint  effect  of  R28.V  Ib.  and  L21£  lb.? 

5.  What  would  be  the  joint  effect  of  each  of  these  pairs  of 
forces : 

(1)  R  481 lb.  and  L  12$  lb.?         (4)  R  23  J  lb.  and  L  68ft  lb.? 

(2)  R  75ft  lb.  and  L  69ft  lb.?     (5)  R  IG-ft  lb.  and  L  81 J  lb.? 

(3)  R  1814  lb.  and  L  129ft  lb.?    (6)  R  87-J-  lb.  and  L  684|  lb.? 

A  force  of  25  T.,  pulling  forward,  may  be  written  F25  T. 
and  a  force  of  18  T.,  pulling  backward,  may  be  written  B  18  T. 
The  joint  effect  of  both  would  be  written  thus :  F  7  T. 

G.  What  is  the  joint  effect  of  the  three  forces:  F 17  T., 
F12  T.  and  B25  T.? 

7.  What  is  the  joint  effect  of  each  of  the  following  sets  of 
forces : 

(1)  F  80|  T.,  B  28ft  T.,  and  B  96f  T.? 

(2)  F494  T.,  B84J  T.,  and  F 248ft  T.? 

(3)  B  68ft  T.,  F  73 j-  T.,  and  B  12f  T.? 

(4)  F28¥3o  T.,  F  184  T.,  and  B  37ft  T.? 

(5)  F  692J  T.,  B  386f  T.,  and  B  307ft  T.? 

(6)  F28zT.,  F3zT.,  andB36.TT.? 

8.  A  force  of  182  lb.  pulling  upward  may  be  written  U  182  lb. 
How  would  you  write  a  force  of  78  lb.  pulling  downward? 

.  What  is  the  joint  effect  of  each  of  the  following  sets  of 
forces: 

(1)  U  8|  lb.,  D  16^  lb.,  and  U3|  lb.? 

(2)  TJ164  lb.,  U28  lb.,  and  D  294  lb.? 

(3)  D  29  oz.,  D  32J  oz.,  and  D  16f  oz.? 

(4)  D  29|  oz.,  U  65£  oz.,  and  D  424  oz.? 

10.  If  we  call  a  force  of  16  lb.  pulling  forward,  or  upward, 
or  eastward,  or  northward,  or  to  the  right,  a  positive  force,  and 
write  it  +16  lb.,  we  should  call  the  same  force  pulling  backward, 
or  downward,  or  westward,  or  southward,  or  to  the  left,  respect- 
ively, a  negative  force.     How  should  we  write  it? 

11.  Give  the  joint  effect,  with  proper  sign  (+  or  — )  of  each 
of  these  sets  of  forces : 

(1)  2  forces,  each  +  8J  lb.,  and  one  force,  — 12|  lb. 

(2)  3  forces,  each  + 124  oz-  j  an(^  ^  forces,  each  —  84  oz. 


COMMON    FRACTIONS 


185 


(3)  5  forces,  each  -  10J  T.,  and  U>  forces,  each  -f -02  T. 

(4)  18  forces  of  -  10T9g-  lb.  each,  and  20  forces  of  +  25J  Ib. 
each. 

12.   Give  the  joint  effect,  with  proper  sign,  of  the  set  of  forces 
in  each  horizontal  line  of  the  table : 


SET 
No. 

No.  OF 
FORCES 

STRENGTH  ob' 
EACH  FORCE 

No.  OF 
FORCES 

STRENGTH  OF 
EACH  FORCE 

JOINT  EFFECT 
WITH  SIGN 

1                 1 

+  25  oz. 

2 

—  16]  07.. 

8 

2 
5 

- 

-  15J3  Hi. 
-  75£  lb. 

1 

8 

—  186f  lb. 

4 

27 

-  28i3-  lb. 

16 

—  624  lb 

5 

16 

-  42  tV  T. 

7 

—  81f  T.' 

6 

9 

-  73|  lb. 

73 

—  9|  lb. 

7 

36 

_  25-7-  lb. 

25 

36^  lb 

8 

3 

4-  x  lb. 

2 

—  xlb. 

9 

12 

-\-xlb. 

7 

—  xlb. 

10 

16 

4-  2x  lb. 

28 

—  xlb. 

11 

16 

-f  5m  lb. 

23 

—  2m  lb. 

12 

25 

+  10m  lb. 

36 

—  12m  lb. 

13 

12 

_L  39  fi  T. 

16 

23«  "j1 

14 

1 

4-  x  ib. 

1 

-ylb.  ' 

15 

3 

_j_  x  lb. 

2 

-ylb. 

16 

8 

h  2x  lb. 

5 

-  2y  lb. 

17 

a 

-xlb. 

a 

—  x  lb. 

18 

a 

-  x  lb. 

a 

—  3x  lb. 

19 

a 

h*lb. 

a 

-ylb. 

§119.  Algebraic  Phrases. 

Letting   x   and   y  denote  any  two   numbers  (x   being  larger 
than  ?/),  represent  in  symbols: 

(1)  Their  sum. 

(2)  Their  difference. 

(3)  Their  product 

(4)  Their  quotient.     (Two  results.) 

(5)  Their  ratio.     It  is  important  to  note  that  quotient  and 
ratio  mean  the  same  thing. 

(6)  The  square  of  the  larger. 

(7)  The  sum  of  their  squares. 

(8)  The  square  of  their  sum  is  written  thus,  (x  +  y)z. 

(9)  The  difference  of  their  squares. 

(10)  The  square  of  their  difference. 


186  RATIONAL    GRAMMAR   SCHOOL   ARITHMETIC 

(11)  The  cube  of  the  smaller  is  written  ?/3,  meaning  y  x  y  x  y. 

(12)  The  sum  of  their  cubes. 

(13)  The  cube  of  their  sum  is  written  thus,  (./:  -f  y)s. 

These  phrases  are  frequently  used  in  algebra,  and  their  meanings 
should  be  clearly  comprehended. 

§120.  Dividing  Lines,  and  Angles. 

PROBLEM    I. — Divide   the   line   AB  into   2  equal   parts,    see 
Problem  VI,  p.  107. 

EXERCISES 

1.  Draw  a  straight  line  with  a  ruler  on  the  blackboard  and 
bisect  it  with  crayon,  string  and  ruler. 

2.  Bisect  each  of  the  3  sides  of  a  triangle  and  connect  each 
mid-point  with  the  opposite  corner  of  the  triangle.     These  lines 
are  the  medians  of  the  triangle.     How  do  they  cross  each  other? 

3.  How  might  a  line  be  divided  into  4  equal  parts  by  repeating 
this  method?     Divide  a  line  into  4  equal  parts  by  this  method. 

PROBLEM  II. — Divide  the  angle  BAD  into  2  equal  parts. 

EXPLANATION.  —  With  any  convenient 
radius  and  with  the  pin  foot  on  A  (vertex) 
draw  arcs  1  and  2  across  AB  and  AD. 

Place  the  pin  foot  on  the  crossing  point  at 
2,  and  with  a  radius  longer  than  half  way  from 
2  to  1  draw  an  arc  3. 

Now  place  the  pin  foot  on  1  and  with  the 
t*  '  'i  *  radius  used  for  arc  3  draw  arc  4  across  arc  3. 

Call  the  point  of  crossing  C. 
Angle  Bisected  With   the   ruler   dra&w   the    bisector    AC 

FIGURE  99  Then  angle  BAG  =  angle  CAD. 

EXERCISES 

1.  Draw  an  angle  on  paper,  or  on  the  blackboard,  and  with 
pencil,  or  crayon  and  string,  bisect  the  angle. 

2.  Draw  a  triangle  and  bisect  each  of  its  3  angles.     How  do 
the  bisectors  of  the  angles  cross  each  other? 

3.  The  line  CD  of  Fig.  45,    p. "  107,  drawn   from   C  to  D, 
is  called   the  perpendicular  bisector  of  the  line  AB.     Draw  a 
triangle  of  3  unequal  sides,  and  then  draw  the  3  perpendicular 
bisectors  of  its  sides.     How  do  these  lines  cross  each  other? 


COMMON    FRACTIONS 


187 


FIGURE  100 


§121.  The  Parallel  Ruler. — A  very  good  parallel  ruler  may  be  made 

by  cutting  out  two  strips  of  cardboard  or  of  very  thin  wood,  exactly 

alike,  as  shown  at  («), 

Fig.  100.  The  numbers 

show  the  widths  and 

the  lengths.     To  use 

the    ruler    place    the 

two  parts  with   their 

long    sides    together, 

the   thin   ends    being 

in  opposite  directions 

as  in  the  cut. 

Another  sort,  which 
the  pupil  may  make 
for  himself  or  pur- 
chase for  a  few  cents, 
is  shown  at  (b) ,  Fig.  100.  The  strips  S  may  be  of  light  cardboard  or 
of  thin  wood  of  the  lengths  and  the  widths  shown  in  the  cut.  The 
outside  edges  of  these  strips  should  be  made  as  smooth  and  as 
straight  as  possible.  The  cross  strips,  which  may  be  narrower  than 
the  strips  jS9  should  be  of  exactly  the  same  length  between  the 
pins  at  (7,  />,  and  at  E,  F.  The  distances  EC  and  FD  between 
the  pins,  should  also  be  made  equal.  Common  pins  may  be  stuck 

through  and  bent  over 

W to  hold  the  strips  to- 

-__ m  gether  at  C9  D,  E,  and 

F. 

The   pupil   should 

provide  himself  with 
a  parallel  ruler  for  the 
following  problems : 

PROBLEM  I. — Draw 
a  line  parallel  to  a 
given  line. 

EXPLANATION.  —  Let 
AB  be  the  given  line. 

Fig.  101  (a)  shows  how 
this  is  done  with  the  first 
kind  of  ruler,  by  holding 
the  part  2  on  the  line, 

A  . — /?     and  sliding  part  1  along, 

FIGURE  101  drawing  the  lines  along 

the  upper  edge  of  part  1. 

Fig.  101  (6)  shows  how  to  solve  the  problem  with  the  second  kind  of 
ruler. 


188 


RATIONAL   GRAMMAR    SCHOOL   ARITHMETIC 


PROBLEM  II.  —  Draw  a  line  parallel  to  a  given  line  and  through 
a  given  point. 

D  EXPLANATION.—  Sup- 

C  -       _  £  _  2)   P?se  AB.  (Fis-  102)  is  the 

given    line,    arid    P  the 
point. 

Hold  one  edge  of  the 
parallel  ruler  along  AB, 
raise  the  other  strip  until 
its  edge  goes  through 
the  point  P,  and  draw  a 
line  CD  along  this  edge. 
CD  is  the  desired  line 


FIGURE  102 


PROBLEM  III.  —  Solve  Problem  II  with  ruler  and  compass. 

EXPLANATION.  —  First  Step:  Place  the  pin  foot  on  the  given  point  P 
and  spread  the  feet  until  the  pencil  foot  reaches  some  point,  as  C,  on  the 
line  AB,  Fig.  103.  Draw  the  arc  C2. 

Second  Step:  Place  the  pin 
foot  on  C  and  using  the  same 
radius  as  before,  draw  arc  Pi, 
cutting  AB  at  D. 

Third  Step  :  Spread  the  feet  of 
the  compass  apart  as  far  as  from  P 
to  D,  and  placing  the  pin  foot  on 
C,  draw  the  short  arc  3.  Connect 
the  crossing  point  E  with  P.  EP 
is  the  desired  parallel  to  AB  through  P. 


FIGURE  103 


EXERCISES 

1.  Draw  a  line  on  paper,  mark  a  point  not  in  the  line  and  with 
a  parallel  ruler  draw  a  line  through  the  point  and  parallel  to  the 
first  line. 

2.  Solve  Exercise  1  on  the  blackboard  with  chalk,  string,  and 
ruler. 

3.  Draw  a  triangle  on  the  blackboard.     Draw  a  line  through 
each  corner  of  this  triangle  and  parallel  to  the  opposite  side. 

PROBLEM  IV. — Divide  a  line  AB  into  3  equal  parts  (or  trisect 
AB). 

EXPLANATION.— Draw  an  indefinite  line 
AC,  making  any  convenient  angle  with  AB. 
Measure  off  3  equal  spaces  from  A  to- 
ward C.    Connect  D  with  B . 

Draw  through  the  points  2  and  1  lines 
parallel  to  DB  (see  Problem  II).    Then  AE  = 

Line  Trisected  ^F  =  FB,  and  E  and  F  are  the  trisection 

FIGURE  104  points. 


COMMON    FRACTIONS  189 

EXERCISES 

1.  Draw  a  line  on  the  blackboard  and  trisect  it. 

NOTE. — Draw  the  parallels  by  the  method  of  Exercise  2  under  Prob- 
lem III  above. 

2.  From  the  suggestion  of  Fig.  105  draw  a  line 
on   the    blackboard  and   divide   it    into   5   equal 

parts.  FIGUBB  105 

3.  How  may  a  line  be  divided  into  7,  or  9,  or  13  equal  parts? 
Draw  a  line  on  the  blackboard  and  divide  it  into  7  equal  parts. 

4.  How  does  the  line  connecting  the  opposite 
corners  of  a  square  divide  the  square? 

5.  Answer  a  question  like  4  for  the  rectangle; 
for  the  parallelogram. 

6.  If,  then,  the  area  of  a  square  equals  the 
FIGURE  ic6           product  of  its  base  and  its  altitude,  what  is  the 

area  of  one  of  the  2  equal  triangles  into  which  a  diagonal  divides 
the  square? 


FIGURE  107 

7.  Answer  similar  questions  for  the  rectangle;  for  the  parallel- 
ogram. 

8.  The  area  (A)  of  a  rectangle  of  base  b  ft.  and  altitude  a  ft.  is 
how  many  square  feet? 

9.  What    is    the   area  of   a   right-angled    triangle    (a    right 
triangle)  of  base  b  in.  and  altitude  a  inches? 

10.  The  base  of  a  parallelogram  is  b  in.  and  the  altitude  is 
a  in. ;  what  is  the  area? 

11.  The  base  of  a  triangle  is  b  and  the  altitude  is  a\  what 
is  the  area? 

12.  The  bases  of  a  rectangle,  of  a  parallelogram  and  of  a  tri- 
angle are  b  in.  and  their  altitudes  are  a  in.     Find  the  ratio  of  the 
area  of  the  rectangle  to  the  area  of  the  parallelogram ;  the  ratio  of 
the  area  of  the  parallelogram  to  the  area  of  the  triangle. 


190 


RATIONAL    GRAMMAR    SCHOOL    ARITHMETIC 


§122.  Uses  of  the  30°  and  the  45°  Triangles. 

PROBLEM  V. — To  make  the  triangles  for  use  in  drawing. 

EXPLANATION. — (a)  Fold  a  piece  of  smooth  heavy  paper,  having  one 
straight  edge  (like  the  piece  shown  in  Fig.  108),  over  a  line  near  the  middle. 
Bring  the  straight  edges  carefully  together  as  shown  in  Fig.  109. 


6in 


FIGURE  108 


FIGURE  109 


Crease  the  paper  smoothly  with  a  ruler  or  a  paper  knife  and  paste  or 
glue  the  two  pieces  together.  When  the  paper  is  dry,  mark  off  distances 
of  4"  from  the  square  corner  on  the  crease  and  on  the  straight  side. 
Connect  the  4  in.  marks  and  cut  the  paper  smoothly  along  the  connecting 
line.  This  will  give  a  triangle  of  the  form  T,  Fig.  110. 

(b)  In  the  same  way,  fold,  crease,  and  paste  another  piece  of  paper 
a  little  larger  than  before.  On  the  straight  side  mark  off  a  distance 
CA  equal  to  3".  With  compasses,  or  with  a  string  or  ruler,  mark  a  point, 
B,  on  the  crease  so  that  AB  equals  6v.  Draw  AB  and  cut  out  the  tri- 
angle S,  Fig.  111. 


3" 


4 

FIGURE  no 


FIGURE  ill 


If  preferred  the  triangles  T  and  S  may  be  made  of  thin  wood. 
EXERCISE.— Place  the  side,  AC,  of  S  against  the  long  side  of  T,  and 


Now 
Simi- 


holding  the  triangles  with  the  left  hand,  draw  a  line  along  AB. 
hold  T,  slide  -S  a  little  (say  i")  and  draw  another  line  along  AB. 
larly,  draw  a  third   line.     Lines  in  such  positions  are  called  parallel 
lines. 


COMMON   FRACTIONS 


191 


PROBLEM  VI. — Through  a  given  point  with  a  triangle  draw  a 
line  parallel  to  a  given  line. 


EXPLANATION.— AB  is 
the  given  line  and  P  is  the 
point  the  parallel  is  to  pass 
through. 

Place  a  ruler  CD,  Fig 
112,  in  such  a  position  that 
when  the  triangle  S  is 
placed  against  it,  one  of 
the  sides  of  S  will  lie 
along  AB.  Press  the  ruler 
against  the  paper  and  hold 
it  with  the  left  hand; 
with  the  right,  slide  the 
triangle  along  the  ruler 
until  its  side  just  touches 
the  point  P.  Draw  line 
FE  through  P  and  along 
the  edge  of  the  triangle. 
FE  is  the  desired  parallel 
line. 


FIGURE  112 


FIGURE  113 
Fig.  113  shows  how  this  problem  is  solved  with  the  triangles  alone. 

EXERCISES 

1.  Solve  the  first  exercise  of  Problem  III  with  the  triangles  8 
and  T  as  shown  in  Fig.  113. 

2.  Draw  a  triangle  with  sides  of  1",  1J",  and  2",  and  through 
each  corner  draw  a  line  parallel  to  the  opposite  side.     Use  the 
ruler  and  the  triangles.     To  draw  the  triangle  see  Problem  IX, 
p.  109. 


192  RATIONAL    GRAMMAR   SCHOOL   ARITHMETIC 

PROBLEM  VII. — At  a  given  point  on  a  line  with  the  triangles 
draw  a  perpendicular  to  the  line. 

EXPLANATION.— Let  AB  be  the  line  and  let  P  be  the  given  point. 

D 

IP 


FIGURE  114  FIGURE  115 

Hold  one  of  the  triangles,  as  T,  in  the  position  shown  in  Fig.  114,  and, 
placing  one  side  of  the  other  triangle,  S,  against  the  upper  side  of  T,  slip 
S  along  until  the  square  corner  comes  to  the  point  P.  Hold  S  firmly  and 
draw  the  line  PD  along  the  side  of  S. 

PD  is  the  required  perpendicular. 

PROBLEM  VIII. — Through  a  given  point  not  in  a  line,  with  the 
triangles  draw  a  perpendicular  to  the  line. 

EXPLANATION.— Let  AB  be  the  line  and  let  P  (Fig.  115)  be  the  given 
point. 

Hold  one  of  the  triangles,  as  T,  so  that  its  side  lies  along  AB  and  slide 
the  other  triangle,  S,  with  its  side  against  the  upper  side  of  T  until  it 
comes  up  to  P.  Then  draw  PD. 

PD  is  the  required  perpendicular. 

EXERCISES 

1.  Draw  a  line,  mark  two  points  on  it  I"  apart  and  .draw  a  per- 
pendicular to  the  line  at  each  of  the  two  points  (by  Problem  VII) . 
Mark  a  point  on  one  of  the  perpendiculars  1"  above  the  given  line 
and  at  this  point  draw  a  third  perpendicular  completing  a  1" 
square. 

2.  Draw  a  triangle  and  through  each  corner  draw  a  perpendic- 
ular to  the  opposite  side  (by  Problem  VIII).     How  do  these  per- 
pendiculars cross? 

3.  Draw  any  circle,  also  a  diameter,  marking  its  ends  A  and  B. 
Through  its  center  draw  a  perpendicular  radius  and  prolong  it, 
making  a  diameter,  CD.     Connect  .4(7,  CB,  BD  and  DA,  forming 
an  inscribed  square. 


COMMON    FRACTIONS 


193 


;  123.   Scale  Drawings  of  Familiar  Objects. 

1.  Notice  the  scale  of  the  drawing  of  the 


Scale?':  6' 


-    5/6   H 


FIGURE  116 


Scale  I": 20' 


playhouse  (Fig.  116),  and  compute  the  lengths 
of  the  following  dimensions : 

(1)  The  width ;  (2)  the  height  of  the  lower 
side;  of  the  higher  side;  (3)  the  rise  (dif- 
ference of  higher  and  lower  sides);  (4)  the 
height  of  the  door;  the  width;  (5)  the  distance 
from  the  right  side  of  the  door  to  the  right 
corner  of  the  house. 

2.  From  the  dimensions  in  Fig.  116,  with  ruler  and  triangles, 
make  a  drawing  of  the  playhouse  to  a  scale  4  times  as  large  as 
that  of  the  drawing  (Fig.  116). 

3.  From  the  scale  of  the  drawing  (Fig. 
117)   find   the  following  dimensions   of   the 
house : 

(1)  The  width;  (2)  the  height  of  the 
eaves;  (3)  the  rise  (aft);  (4)  the  width  and 
the  height  of  the  door;  (5)  the  width  and 
the  height  of  the  window.  FIGURE  m 

4.  Find  the  area  of  the  end  of  the  house,  including  the  gable 
and  excluding  the  areas  of  the  door  and  the  window. 

5.  Make  an  enlarged  drawing  of  the  house  to  a  scale  8  times 
as  large  as  the  scale  of  the  drawing  (Fig.  117). 

6.  From    the   scale   of    Fig.    118, 
give  the  length  and  width  of  the  door ; 
the  length  and  the  width  of  the  panel. 
(Find  the  length  by  measurement). 

7.  Similarly,  -give  the  length   and 
the  width  of  the  door  B,  also  the  length 
and   the  width  of   the  upper   panels, 
and  the  widths  of  the  strips  enclosing 
the  panels. 

8.  Make  an   enlarged   drawing   of 
each  of  the  doors  to  a  scale  5  times  as  large  as  the  scale  of  the 
drawings  (Fig.  118). 


/o 

Xo 

« 

A 

i/" 

, 

Panel 

t, 

^ 

__ 

R 

7° 

e 

7/ao 

nn 

"/20 

'/ao 

5  c  ale  1":  60" 

FIGURE  118 

194 


RATIONAL   GRAMMAR    SCHOOL    ARITHMETIC 


9.  How  long  are  the  cross-pieces  and  the  strings  of  the  kite 
represented  by  the  drawing  (Fig.  119)? 

10.  Make  an  enlarged  drawing  of  the  kite  to  a 
scale  8  times  as  large  as  that  of  Fig.  119. 

11.  Notice  that  the  horizontal  piece  divides  the 
surface  of  the  kite  into  two  trapezoids.     The  alti- 
tude of  the  upper  trapezoid  in  the  drawing   (Fig.         sCa/e  r<54" 
119)  is  £J",  and  that  of  the  lower  trapezoid  is  -J-J".       FIGURE  119 
How    many  square   feet  of   paper  are  needed  to  cover  the  kite? 
(Allow  144  sq.  in.  for  folding  and  pasting  over  strings.) 

12.  What   is  the  length   of   each  leg  of    the 
chair  shown   in   the  drawing  (Fig.  120)?     What 
is   the  height  of  the   back?    the  depth   of    the 
seat  to  the  extreme  rear? 

13.  Make  a  drawing  of  the  chair  to  a  scale  5 
times  as  large  as  that  of  Fig.  120. 

14.  Find  from  the  drawing  of  the  gate  (Fig. 
121),  (1)  the  height  of  the  high  end  of  the  gate; 
(2)  of  the  lower  end;  (3)  the 

length  of  the  long  brace; 
(4)  the  length  of  the  gate ;  (5)  the  width  of 
the  strips. 

15.  Make  a  drawing  of   the   gate   to   a 
scale  8  times  as  large  as  that  of  Fig.  121. 

16.  Fig.    122  is  a  scale  drawing  of   the  FIGURE  121 

jj»  side  and  the  end  views  of  a  large  book.     How  long 

is  the  book?  how  wide?  how  thick? 

17.  Make  an  enlarged  drawing  of  the  book  to  a 
scale  4  times  as  large  as  that  of  Fig.  122. 

18. 'From  your  own  measurements  make  a 
drawing,  to  any  convenient  scale,  of  a  thick  object, 
as  a  block,  a  brick,  a  crayon-box,  showing  two 
views  as  in  Fig.  122. 

19.  Make  a  scale  drawing  from  your  own 
measures  of  a  desk,  table,  bookcase,  or  other 
object  in  your  schoolroom,  showing  three  different  views  (top, 
side,  and  edge  views)  of  it. 


Jcafe  r:40" 
FIGURE  120 


END    VIEW 
Scale  I":  16' 

FIGURE  122 


COMMON    FRACTIONS 


195 


§124.  Schoolhouse  and  Grounds. 

1.  Using  a  foot  rule,  graduated  to  16ths  of  an  inch,  and 
regarding  the  scale  of  the  drawing  (Fig.  123),  find  the  width  of  the 
grounds;  the  length; 
the  area  in  square 
rods.  (30^  sq.  yd.= 
1  square  rod.) 

2.  Find  the  length 
of     the     field;     the 
width;    the   area  in 
square  rods. 

3.  Find  the  length 
and  the  width  of  the 
school  yard ;  the  area 
in  square  rods. 

4.  Find  the  length 
and  the  width  of  the 
schoolhouse ;  the  area, 
in  square  yards,  cov- 
ered by  it. 

5.  How  far  is  the 
front   door   of   the 
schoolhouse  from 
the  front  fence?  from 

the  west  front  gate?  from  the  sand  pile?  from  the  tree? 

G.  How  far  is  it  from  the  back  door  to  the  east  flower  bed?  to 
the  back  fence?  to  the  west  fence?  to  the  coal  shed?  to  the  north- 
east corner  of  the  school  yard?  to  the  south  end  of  the  pond?  to 
the  hill?  to  the  nearest  point  on  the  creek  bank?  to  the  foot 
bridge  (F)? 

7.  How  wide  is  the  south  road?  the  creek?  the  branch? 

8.  How  many  square  rods  in  the  south  road  in  front  of  the 
grounds?  in  the  crossing  of  the  roads? 

9.  How  many  square  rods  in  the  meadow?  in  the  grove?  in  the 
pasture? 

10.  How  many  square  rods  are  covered  by  the  creek  and  the 
branch  together,  within  the  fence  lines? 


ROAD 


Scale  I":  160' 
FIGURE  123 


196  RATIONAL    GRAMMAR    SCHOOL    ARITHMETIC 

11.  How  many  rods  of  fence  will  be  needed  to  enclose  the 
grounds  and  to  run  along  the  lines  indicated? 

12.  From  your  own  measurements  make  a  similar  drawing,  to 
a  convenient  scale,  of  some  tract  of  ground  (school  yard  or  field) 
near  your  schoolhouse.     Locate  any  fixed  objects  on  your  tract 
in  their  proper  places  on  the  drawing  by  measuring  their  shortest 
distance  from  a  fence  and  from  a  corner. 

§125.  Proportion.  ORAL  WORK 

1.  Compare  the  ratio  3  :  2   with  the  ratio  6  :  4.      What   do 
you  find? 

2.  Compare   the   ratio   8  d.  :  2d.    with    the    ratio    16    men: 
4  men. 

3.  Compare  the  ratio  10  ft.  :  800  ft.  with  the  ratio  100  mi. : 
8000  miles. 

4.  A  proportion  is  an  equation  of  ratios.     Thus  6  :  12  =  3  :  6 
and  T6^  =  £  are  two  different  ways  of  writing  the  proportion. 

DEFINITIONS. — The  first,  second,  third,  and  fourth  numbers  of  the  pro- 
portion are  called  the  first ,  second,  third,  and  fourth  terms  of  the  propor- 
tion. The  first  and  fourth  terms  are  called  the  extremes,  and  the  second 
and  third  terms  are  the  means.  The  first  two  terms  are  the  first  couplet, 
the  third  and  fourth  terms  are  the  second  couplet. 

5.  In  6  :  12  =  3  :  6,  to  what   is  the  product  of  the  extremes 
equal?  the  product  of  the  means? 

6.  Answer  the  same  questions  for  3  :  5  =  12  :  20. 

7.  Which  of  these  pairs  of  ratios  may  form  proportions: 

4  :  9  and    8  :  18?          6  :  11  and  18  :  33?         1:3  and  8  :  21? 
3  :  7  and  12  :  28?          6  :  11  and  12  :  22?         f  and  TV^? 

1  and  A?  ±  and  -1?  2.  and  ^L? 

MM  ax  ex  n  6n 

8.  Is  4  =  ii  a  proportion?    >. 

Multiply  both  sides  by  21  and  we  have  4  x  21  =  15. 

Now  multiply  both  sides  of  this  equation  by  7  and  we  have 
5x21  =  7x15. 

What  were  the  terms  5  and  21  in  the  proportion  called?  What 
were  the  7  and  15  in  the  proportion  called? 


COMMON    FRACTIONS  197 

9.  Compare  the  product  of  the  1st  and  the  4th  numbers  in 
these  proportions  with  the  product  of  the  3d  and  the  3d 
terms : 

(i)  *  =  if  ;  (5)  A  =  M ; 

00  A  -  * ?  ;  (6)  |  =  &;    • 

(3)  tt  -  B  ;          (?)  T  =  His 

(4)  {  .iff.;  (8)     f-tflf. 

Can  you  state  the  principle  problems  8  and  9  illustrate? 

PRINCIPLE. — In  a  proportion  the  product  of  the  means  equals 
the  product  of  the  extremes. 

WRITTEN   WORK 

What  must  x  be  in  each  of  these  expressions  to  give  a  propor- 
tion? Solution  of  first  equation:  2x=  15,  or  x  =  7.5. 

1.  2:3  =  5:z  6.  a:b  =  2a:x 

2.  4:3  =  8:2;  7.  18:z  =  54:18 

3.  6:z  =  9:27  8.  4:6=    z:9 

4.  7:2  =  *:14  9.  «  =  ^ 

5.  x:  6  =  8:12  10.  ^  -  ^ 

be  o 

§126.  Practical  Applications. 

1.  If  £•  of  a  ton  of  coal  is  worth  $4.81,  what  are  12  T.  worth 
at  the  same  rate? 

NOTE. — Call  the  unknown  term  x.  Write  the  proportion,  using  x, 
and  then  use  the  Principle,  §125,  to  find  x. 

2.  A  spelling  class  of  52  pupils  writes  a  total  of  1352  words; 
at  the  same  rate  how  large  a  class  will  write  a  total  of  728  words? 

3.  A  girl  jumping  a  rope  makes  441  skips  in  3  min.     How 
long  w,ill  it  take  her  to  make  392  skips  at  the  same  rate? 

4.  If  the  average  column  in  a  newspaper  contains  1600  words, 
how  much  should  a  writer  receive  for  700  words  at  the  rate  of 
$5  per  column? 

5.  In  going  10,725  ft.  the  front  wheel  of  a  bicycle  revolves 
1430  times.     How  far  would  it  go  in  making  1001  revolutions?' 


198 


RATIONAL   GRAMMAR    SCHOOL   ARITHMETIC 


6. 

it  tick 

7. 


A 

in 
If 


BEECH   BROC 


clock    ticks  7  times   in  5  sec.,  how  many  times   does 

2  da.,  6  hours? 

60  A.  cost  $3000,  how  many  acres  will  cost  $2450? 

8.  The  line  BC  (Fig.   124) 
represents  20  rods.     FE  is  twice 
as  long,  AF  four  times  as  long, 
CD  and  ED  five  times  as  long, 
and    AB    six    times    as    long. 
Find  the  length  of  each  side. 

9.  Find  the  value  of  x  in  the 
following  proportions: 

BC  :  FE  —  AB  :  (x); 
BC  :  (x)  =  FE  :  CD; 
AF  :  FE  =  (x)  :  BC. 


FIGURE  124 


-T-— ""  4 

3h 1 


FIGURE  125 


10.  When  a  foot  rule  is  held 
2£  ft.   (arm's  length)  in   front 

of  the  face  as  shown  in  Figure  125,  7   in.  on  the  ruler   seems 

just  to  cover  the  edge  of  a  door  7  ft.  high.     If  the  ruler  is  held 

parallel    to   the   edge   of    the 

door,  how  far  is  the  eye  from 

the  door? 

11.  If  the  shadow  cast  by  a 

4-ft.  stake  is  5  ft.  long,  how  high 

is  a  tree  which  casts  a  shadow  50  ft.  long  on  the  same  day  and  hour? 

12.  A  woodman  desires  to  fell  only 
such  trees  as  will  furnish  two  10-ft. 
cuts  between  the  stump  and  the  first 
limb.  He  wishes  to  allow  4  ft.  for 
height  of  stump  and  waste  in  cutting 
at  the  bottom  and  top  ends.  To  test 
a  standing  tree,  he  places  a  4-ft. 
stick  QR  vertically  in  the  ground  33  ft. 
from  the  tree,  lies  down  on  his  back 
with  his  feet  against  the  stake,  and 
sights  over  the  top  of  the  stake  to  the 

first  limb  B.     The  distance  from  his  eye  P  to  the  soles  of  his 

feet  Q  being  5|  ft.,  should  he  fell  the  tree? 


FIGURE  126 


13.  If 
KD  (Fig.  127)? 


COMMON    FRACTIONS  199 

rd.,  AK=  80  rd.,  and  BC=  56  rd.,  how  long  is 


14.  Make  measures  on  objects  in  your  vicinity  and  solve  such 
problems  as  are  suggested  by  those  given. 

15.  How  long  is  x  (Fig.  128)? 

16.  The    thumb   is   held   2   in. 
from  the  end  of  the  pencil,  and  the 
pencil  is  held  2  ft.  in  front  of  the 

eye  and  parallel  to  the  line  to  be  measured.     When  the  end  of 
the  pencil  is  sighted  into  line  with  the  corner  5  of  the  table,  the 


FIGURE  128 


FIGURE  129 

end  of  the  thumb  is  in  line  with  the  corner  a.     How  long  is  the 

end,  at),  of  the  table,  if  the  eye  is  36  ft.  from  the  table? 

17.  The  eye  sights  past  a 
point  A  over  the  point  P  on 
the  fence  (Fig.  130),  and  sees 
the  upper  edge  of  the  moon  in 
line  with  A  and  P.  Then 
moving  the  eye  2£  in.  up  the 
stake,  the  lower  edge  of  the 
moon  is  in  line  with  B  and  P. 
If  the  stake  is  20  ft.  from  the 

fence  and   the  moon  is   240,000  miles  away,  how  long  is  the 

moon's  diameter  in  miles? 

18.  Make  measures  like  these  yourself. 


200  RATIONAL   GRAMMAR   SCHOOL   ARITHMETIC 

±9.   The  rays  of  light  from  the  sun  passing  through  a  pin-hole 
in  the  screen  at  H  (Fig.  131)  give  a  small  circular  image  of  the 


FIGURE  131 

sun  at  /.      With  distances  and  diameter  of  image  as  shown  in 
the  cut,  what  is  the  diameter  of  the  sun? 

20.  A  small  hole,  H,  in  a  window  screen,  gave  a  round  image 
of  the  sun,  1.1  in.  in  diameter  on  a  sheet  of  paper  held  10  ft.  from 
the  pin-hole.      If  the  sun  is  93,000,000  mi.   away,  what  is  the 
sun's  diameter? 

21.  Solve  this  problem  from  your  own  measures. 


DECIMAL  FRACTIONS 

§127.  Notation  of  Decimals.    ORAL  WORK 

1.  In  $1111  what  does  the  1st  1  on  the  right  stand  for?  the 
2d?  the  3d?  the  4th? 

2.  In  $6666  what  does  the  1st  6  on  the  left  denote?'  the  2d? 
the  3d?  the  4th? 

3.  In  $372.68  what  is  the  unit  of  the  3  (Am.  $100)?  of  the  7? 
of  the  2?  the  6?  the  8? 

4.  In  $5555  how  does  the  number  denoted  by  each  5  compare 
with  the  number  denoted  by  the  5  to  its  left?  to  its  right? 

5.  How  do  the  units  of  the  places  of  a  number  change  as  we 
pass  through  the  number  from  left  to  right? 

6.  Which  unit  is  the  fundamental  unit  out  of  which  the  other 
units  are  made?     How   is  the   place  of  this   fundamental   unit 
indicated  in  $675.28? 

DEFINITION. — A  dot,  called  the  decimal  point,  or  point,  is  used  to  show 
the  units'  digit.  The  point  always  stands  just  to  the  right  of  the  units' 
digit  or  place. 


DECIMAL    FRACTIONS  201 

7.  If  the  law  of  problem  5  holds  in  444.444,  the  unit  of  the 
1st  4  to  the  right  of  the  point  equals  what  part  of  the  unit  of  the 
1st  4  to  the  left  of  the  point?     The  unit  of  the  2d  4  to  the  right 
equals  what  part  of  the  unit  of  the  1st  4  to  the  right? 

8.  In  any  number  what  part  of  the  unit  of  the  1st  place  to 
the  left  of  the  point  equals  the  unit  of  the  1st  place  to  the  right? 
of  the  2d  place  to  the  right?  of  the  3d?  of  the  4th? 

DEFINITION.— The  unit  of  the  1st  place,  or  digit,  to  the  right  is  called 
the  tenth;  of  the  2d  place,  or  digit,  the  hundredth;  of  the  3d,  the 
thousandth;  of  the  4th,  the  ten-thousandth  and  so  on. 

§128.  Numeration  of  Decimals. 

1.  The  number  444.444  might  be  read,  "4  hundreds,  4  tens, 
4  units  and  4  tenths,  4  hundredths,  4  thousandths";  but  it  is 
simpler  to  read  it,  "Four  hundred  forty -four  and  four  hundred 
forty-four  thousandths.11    Read  234.234  both  ways.    Which  is  the 
shorter? 

2.  The  latter  mode  of  reading  is  the  one  used  in  practice.     It 
may  be  stated  thus :  Read  the  integral  (whole)  part  of  the  number, 
then  read  the  part  of  the  number  to  the  right  of  the  point  just 
as  though  it  were  a  whole  number  standing  to  the  left  of  the 
point,  then  pronounce  tke  name  of  the  unit  of  the  last  digit  on 
the  extreme  right.     The  word  "and"  must  be  pronounced  only  at 
the  decimal  point.     Read  675.328;  236.89;  7.65;  43.6587. 

3.  Read  the  following 

(1)  .1;         .01;  .001;     .0101;         1.1;  10.1; 

(2)  .6;       6.06;       66.6;         .0606;     600.06;       6000.006; 

(3)  .60;     7.070;       8.50;       .0600;       87.087;         10.0008. 

4.  What  is  the  ratio  of  5  to  .5;    of  .5  to  .05;   of  .5  to  .005; 
of  55  to  5.5;  of  55  to  .55? 

5.  Find  the  ratio  of  875  to  87.5;    of  87.5  to  8.75;    of  875  to 
8.75;  of  875  to  .875. 

6.  How   is  a  number  affected  by  moving  the  decimal  point 
1   place   toward   the    left    (see    problems   4   and   5)?    2   places? 
3  places?  6  places? 

7.  What  is  a  quick  way  of  dividing  any  number  by  10?   100? 
1000? 


202  RATIONAL   GKAMMAR   SCHOOL   AEITHMETIC 

8.  Express  the  ratio  of  2.358  to  23.58;  to  235.8;  to  2358; 
to  23,580;  to  .2358;  to  .02358. 

QUERY.— Where  is  the  point  supposed  to  be  in  2358?  When 
the  decimal  point  is  not  written,  where  is  it  supposed  to  be? 

9.  How  is  a  number  changed  by  moving  the  decimal  point 
1  place  to  the  right?  2  places?  3  places?  6  places? 

10.  Make  a  rule  for  quickly  multiplying  any  number  by  10; 
by  100;  by  1000;  by  1  followed  by  any  number  of  zeros. 

11.  Kef  erring  to  problem  4,  can  you  tell  what  effect  is  produced 
in  such   a  number  as  .683  or  .492  by  writing  a  zero  between  the 
decimal  point  and  its  first  digit?   2  zeros?  any  number  of  zeros? 

12.  Compare   .5   with   each   of    these   numbers:    .50;    .500; 
.50000.     What  is  the  effect  of  writing  any  number  of  zeros  to 
the  right  of  the  last  digit  of  a  decimal? 

DEFINITIONS. — A  decimal  fraction,  or  decimal,  is  a  fraction  whose 
denominator  is  10,  100,  1000,  or  some  power  of  ten,  in  which  the  denom- 
inator is  not  written  but  is  indicated  by  the  position  of  the  decimal  point. 

A  power  of  10  is  a  number  obtained  by  using  10  as  a  factor  any 
number  of  times. 

§129.  To  Reduce  a  Decimal  to  a  Common  Fraction. 

The  fractions  we  have  studied  whose  denominators  are  actually 
written,  are  called  Common  Fractions. 

1.  Express  the  following  decimals  as  common  fractions: 

.5;  .50;  .500;  .4;  .40;  .400;  .6;  .60;  .25;  .250;  .125;  .375; 
.625;  .875. 

2.  Eeduce  the  results  of  problem  1  to  their  lowest  terms. 

3.  Write  these  mixed  decimals  as  improper  fractions : 

1.5;  2.50;  2.75;  10.4;  12.5;  6.75;  18.25. 

4.  Eeduce  these  improper  fractions  to  their  lowest  terms. 

5.  After   dropping    the   decimal    point    from    the   following 
decimals,  what  numbers  must  be  written  beneath  them  to  express 
them  as  common  fractions : 

.6;  .67;  .625;  .875;  .1275;  12.75;  25.786;  33.333;  6.6666? 

6.  Make  a  rule  for  expressing  any  decimal  as  a  common  frac- 
tion in  its  lowest  terms. 


DECIMAL   FRACTIONS  203 

7.  How  may  2-J-  tenths  be  expressed  as  a  decimal?  62£  hun- 
dredths?  33  J  hundred  ths?  14  and  666|  thousandths? 

A  pure  decimal  is  a  decimal  whose  value  is  less  than  1;  as, 
38  thousandths,  66|  hundredths.  A  mixed  decimal  is  a  decimal 
whose  value  is  greater  than  1:  as,  3.58  or  2.87^. 

8.  Read  and  give  the  meaning  of  these  mixed  decimals  : 
.12i;  3.33£;  18.66f;  2.1^;  .Of;  .04f  ;  .OOf;  36.000J. 

NOTE.  —  Numbers  expressed  in  both  decimals  and  common  fractions 
are  called  complex  decimals.  A  simple  decimal  is  expressed  without  the 
use  of  common  fractions. 

To  reduce  such  an  expression  as  3.44f  to  a  mixed  number  pro- 
ceed thus: 


3445      _s_ya      2413      0313 

SOLUTION.-3.44?  =  -Z  =  _,_  =  __  =  3^. 

445  313  ai_3.    vx  7  Q-jQ 

Or,  thus:  3.44?  =  3^  =  3^=  3^  *  J=8«« 

9.  Reduce  the  decimals  of  problem  8  to  mixed  numbers  or 
common  fractions. 

10.  Make  a  rule  for  expressing  any  mixed  decimal  as  a  common 
fraction,  or  a  mixed  number. 

PRINCIPLE  I.  —  Any  decimal  may  be  expressed  as  a  common 
fraction  in  its  lowest  terms  or  as  a  mixed  number,  by  dropping  the 
decimal  point,  iuriting_  the  denominator,  and  reducing  the  resulting 
common  fraction  to  its  lowest  terms. 
§130.  Rain  and  Snowfall,  or  Precipitation  —  (ADDITION). 

DEFINITION.  —  1  in.  of  rainfall  means  a  fall  of  1  cu.  in.  of  water  on 
each  square  inch  of  surface  of  the  ground.  (Review  §51,  pp.  68-9). 

1.  The  following  quantities  of  rain  fell  from  week  to  week 
during  May,  1902,  in  Chicago;  find  the  total  rainfall  for  the 
month:  First  week,  1.09  in.;  second  week,  1.11  in.;  third  week, 
.45  in.,  and  fourth  week,  2.43  inches. 

SOLUTION.— 

CONVENIENT  FORM  ExpLANATiON.-It  is  convenient  to  write  the 

}n-         addends  in  a  column  so  that  the  units  digits  are  all 

in  the  same  vertical  column.     Then  begin  on  the 

.45  m.         right  and  add  as  with  whole  numbers.     In  the  sum 

2.  43  in.         the  point  should  stand  directly  under  the  points  in 

Ans.    5.08  in.         the  addends. 

DEFINITION.  —  Finding  the  sum  of  decimal  numbers  is  called  addition 
of  decimals. 


204  RATIONAL   GRAMMAR   SCHOOL   ARITHMETIC 

2.   The  following  numbers  denote  the  monthly  precipitations 
for  1902;  what  was  the  total  precipitation  for  the  year? 

.66    1.53    4.16    2.26    5.08    6.45    425    1.44    4.83    1.45    2.03    1.90 

5.  Without  rewriting  the  numbers,  find  the  total  yearly  pre- 
cipitation for  the  years  1891-1901  from  these  recorded  data: 


JAN. 

1.99 
1.99 
2.08 
1.55 
2.15 

1.12 
4.53 
3.54 
.58 
1.21 
1.15 

FEB. 

1.95 
1.57 

2.44 
2.13 
1.60 

3.48 
2.22 
2.59 
1.60 
3.52 
2.05 

MAR. 

2.13 
2.21 
1.69 
2.66 
1.32 

1.26 
3.56 
4.60 
2.11 
1.58 
3.38 

APR. 

2.14 

2.17 
4.16 
».  65 
36 

2.79 
2.23 
.76 
.14 
1.02 
.33 

MAY 

2.09 
6.77 
1.93 
3.35 
1.99 

4.16 

.84 
2.23 
4.35 
3.59 

2.18 

JUNE 

2.42 
10.58 
3.59 
1.96 
1.79 

2.82 
3.60 
5.30 
2.71 
2.06 
2.42 

JULY 

2.47 
2.23 
3.08 
.60 
2.43 

3.61 
1.47 
1.94 
6.66 
4.64 
4.25 

AUG. 

4.52 
1.85 
.18 
.60 
6.49 

3.52 
1.70 
3.03 
.91 

4.24 
2.00 

SEPT. 

.32 
1.34 
1.98 

8.28 
1.89 

6.70 
.84 
3.16 
2.39 
1.56 
2.92 

OCT. 

.36 
1.54 
1.75 
.85 
.51 

1.36 
.18 
3.26 
2.09 
1.35 
1.29 

Nov. 

3.83 
2.68 
2.45 
1.18 
5.60 

2.16 
3.06 
2.25 
1.14 
3.30 
.85 

DEC.  TOTAL 

1.32 
1.63 
2.14 
1.66 
6.76 

.16 
1.62 
1.11 
6.81 
.58 
1.70 

4.   Foot  and  average  the  vertical  columns  and  tell  what  the 
footings  and  averages  mean. 

§131.  Other  Applications. 

1.  A  coal  dealer  received  8  carloads  of  coal  of  the  following 
tonnages:  24.6,  28.785,  31.25,  24.95,  31.8,  25.125,  28,  and  29.25. 
What  was  the  total  tonnage? 

2.  A  farm  was  divided  by  its  owner  into  lots  of  the  following 
acreages:    32.874,    7.124,    68.334,    11.66J,    16.28f,  and   21.13J. 
What  was  the  total  area  of  the  farm? 

3.  The  following  numbers  represent  in  thousands  of  feet  a 
lumber  dealer's  sales  in  1  da. :  6.865,  24.245,  16.398.  12.28,  18.2, 
6.395,  24,  and  18.967.     What  was  the  total  sales  for  the  day? 

4.  A  quantity  of  soil  contained  .125  Ib.  gravel;  .268  Ib.  coarse 
sand;  .175  Ib.  fine  sand;  .0374  Ib.  organic  matter;  .214f  Ib.  clay, 
and  .275  Ib.  water.     What  was  the  total  weight  of  the  soil? 

5.  During  8  hr.  a  freight  train  made  the  following  mileages: 
32.15,  28.375,  15.687,  20.2,  15.63,   17.5,  8.95,  and  21.3.     Sow 
far  did  the  train  run  during  the  8  hours? 

6.  Following  are  the  weights  in  grains  of  the  U.  S.  coins: 
10-piece,   48;    5^piece,  73.166|;    dime,  38.5834;   quarter  dollar, 


DECIMAL    FRACTIONS 


205 


96.45;  half  dollar,  192.9;  dollar,  412£;  quarter  eagle,  64£;  half 
eagle,  129;  eagle,  258;  double  eagle,  516.  Find  the  total  weight 
of  all. 

§132.  Nature  Study — (SUBTRACTION). 

1.  54  cu.  in.  of  soil  in  its  natural  state  weighed  1.94  Ib. 
After  being  thoroughly  dried  it  weighed  1.459  Ib.  How  much 
moisture  passed  off  in  drying? 

EXPLANATION.— We  have  seen  that  1.94  may  be 
written  1.940.  For  convenience  write  the  numbers 
so  that  units'  digits  stand  in  the  same  column.  Be- 
ginning on  the  right  subtract  as  though  the  numbers 
were  whole  numbers.  In  the  result  the  point  should 
stand  directly  under  the  points  in  the  minuend  and 
subtrahend. 


SOLUTION. — 
CONVENIENT  FORM 

1.940  Ib. 
1.459  Ib. 


Ans.  .481  Ib. 


DEFINITION. — Finding  the  difference  of  decimal  numbers  is  called 
subtraction  of  decimals. 

2.  After  drying,  the  same  soil  occupied  only  37.125  cu.  in. 
How  much  did  it  contract  in  bulk  in  drying? 

3.  100   green   oak  leaves  weighed  .22  Ib.      After  thorough 
drying  they  weighed  .087  Ib.     What. was  the  weight   of  water 
contained  in  the  green  leaves? 

4.  The  organic  matter  in  the  leaves  was  then  driven  off  by 
burning  the  dry  leaves.     The  ash  weighed  .0053  Ib.     How  much 
organic  matter  did  the  100  leaves  contain? 

5.  The  corresponding  numbers  for  100  green  elm  leaves  were : 
Weight  of  green  leaves,  .132  Ib. ;  weight  of  dry  leaves,  .0345  Ib. ; 
weight  of  ash,  .0035  Ib.     Answer  questions  like  3  and  4  for  these 
leaves. 

6.  Answer  similar  questions  for  these  leaves: 


KIND  or  LEAVES 

WEIGHT  IN  POUNDS 

FRESH,  GREEN 

DRY 

ASH 

50  Poplar  leaves  

.132 
.099 

.043 
.044 

.0043 
.0026 

35  Compound  ash  leaves. 

7.  1.135  Ib.  of  dry  beans,  soaked  for  24  hr.,  weighed  2.212 
Ib.     What  was  the  amount  of  water  taken  up  by  the  beans? 


206 


RATIONAL    GRAMMAR    SCHOOL    ARITHMETIC 


8.  The  dry  beans  occupied  34.875  cu.  in.  and  the  soaked 
beans  85.5  cu.  in.  How  much  did  the  beans  increase  in 
bulk? 


§183.  Stature  and  Weight  of  Persons. 

The  following  table  contains  the  average  heights  and  weights  of 
boys,  girls,  men,  and  women  for  the  ages  indicated  by  the  numbers 
in  the  first  column.  Heights  are  given  in  feet  and  weights  in 
pounds. 


AGE 
YR. 

HEIGHTS 

Diff. 

GROWTH  IN 
HEIGHT 

WEIGHTS 

Diff. 

GROWTH  IN 
WEIGHT 

Males 

Females 

Males 

Females 

Males 

Females 

Males 

Females 

2 

2.60 

2.56 

25.01 

23.53 

4 

3.04 

3.00 

31.38 

28.67 

6 

3.44 

3.38 

38.80 

35.29 

9 

4.00 

3.92 

49.95 

47.10 

11 

4.36 

4.26 

59.77 

56.57 

13 

4.72 

4.60 

75.81 

72.65 

15 

5.07 

4.92 

96.40 

89.04 

17 

536 

5.10 

116.56 

104.43 

18 

5.44 

5.13 

127.59 

112.55 

20 

5.49 

5.16 

132.46 

115.30 

30 

5.52 

5.18 

140.38 

119.82 

40 

5.52 

5.18 

140.42 

121.81 

50 

5.49 

5.04 

139.96 

123.86 

60 

5.38 

4.97 

136.07 

119.76 

70 

5.32 

4.97 

131.27 

113.60 

80 

5.29 

4.94 

127.54 

108.80 

90 

5.29 

4.94 

127.54 

108.81 

1.  Fill  out  the  vacant  columns  headed  difference  (Diff.) ;  the 
first  by  subtracting  the  height  of  the  females  from  that  of  the 
males  of  the  same  age,  and  the  second  by  subtracting  the  weights 
of  females  from  those  of  males  of  the  same  age.     Subtract  without 
rewriting  the  numbers. 

2.  There  are  4  vacant  columns  headed  growth;  two  for  growth 
in  height  and  two  for  growth  in  weight.     Fill  out  the  first  of 
these  columns  from  the  column  of  heights  of  males  by  subtracting 
each  number  of  this  column  from  the  one  next  below  it.     Tell 


DECIMAL   FRACTIONS 


207 


what  the  difference  means.     Fill  out  the  second  similarly  from 
the  column  of  heights  of  females. 

3.  In  a  similar  way  fill   out  the  last  two  columns  from  the 
columns  of  weights  of  males  and  weights  of  females.     Tell  what 
these  differences  mean. 

4.  At  what  age  are  boys  growing  most  rapidly  in  height?  in 
weight?     At  what  age  do  men  begin  to  decrease  in  height?  in 
weight? 

5.  Answer  questions  similar  to  4  for  females. 

6.  Compare  your  own  height  and  weight  with  the  numbers  of 
this  table,  for  your  age. 


NOTE. — If  your  age  is  not  in  the  table,  it  will  bs  between  two  _„„ 
given  there.  Use  the  mean  of  the  numbers  for  these  two  ages  for  your 
comparison. 

7.  The  following  table  contains  the  stature  in  feet  of  children 
of  Manchester  and  of  Stockport,  (1)  who  are  working  in  factories, 
and  (2)  who  are  not  working  in  factories.  Without  rewriting  the 
numbers,  fill  out  the  vacant  columns  of  differences  between  the 
heights  of  the  two  classes  for  both  boys  and  girls.  What  effect, 
if  any,  of  such  work  can  you  detect  on  the  growth  of  the  boy  or 
girl? 

NOTE. — When  the  boy  or  girl  not  working  in  factories  is  taller  than 
the  one  working  in  factories,  mark  the  difference  with  a  plus  (-)-)  sign 
before  it.  In  the  opposite  case  mark  the  difference  with  the  minus  ( - ) 
sign  before  it. 


Bo 

YS 

Gil 

ILS 

AGES 

Working 
in  Factories 

Not  Working 
in  Factories 

DlFF. 

Working 
in  Factories 

Not  Working 
in  Factories 

DlFF 

9  years.. 

4.009 

4.045 

3.996 

4036 

10 

4.167 

4.219 

4.134 

4.114 

11 

4.272 

4.252 

4.261 

4.341 

12 

4.446 

4.413 

4.475 

4.472 

13 

4.537 

4.580 

4.636 

4.590 

14 

4.715 

4.725 

4.813 

4.852 

15 

4.971 

4.836 

4.875 

4.928 

16 

5.134 

5.266 

4.990 

5.003 

17 

5  223 

5.338 

5.039 

5.059 

18 

5.276 

5.825 

5226 

5.397 

RATIONAL   GRAMMAR    SCHOOL    ARITHMETIC 

§134.  Pointing  the  Product  of  Decimals — (MULTIPLICATION). 

ORAL    WORK 

1.  What  is  the  relation  between  the  following  pairs  of  num- 
bers? 

(1)  25  and  2.5         (4)   1.28  and  12.8         (7)  2847   and  28.47 

(2)  25  and  .25         (5)  1.28  and    128         (8)   284.7  and  2.847 

(3)  75  and  7.5         (6)  47.8  and  4.78         (9)  28.47  and  .2847 

2.  The  product  37  x  25  equals  how  many  times  the  product 
37x2.5? 

3.  The  product  684  x  7.5  equals  what  part  of  the  product 
684x75? 

4.  What  is  the  product  684  x  75?     What,  then,  is  the  product 
684x7.5?    What  is  the  product  684  x.  75?  684  x. 075?  68.4  x. 075? 
6.84  x. 075?  . 684  x. 075? 

DEFINITION. — By  the  number  of  decimal  places  of  a  number  is  meant 
the  number  of  digits  (zero  included)  on  the  right  of  the  decimal  point. 
ILLUSTRATION. — In  2.005  there  are  3  decimal  places. 

WRITTEN    WORK 

1.  Find  the  product  4862  x  784,  and  from  it  write  the  follow- 
ing products : 

486.2x784;  48.62x784;  48.62  x  78.4;  4.862  x  7.84;  4862  x  .784. 

2.  How  many  decimal  places  are  there  in  486.2?  in  48.62? 
4.862?  10.03?  3.0060?  .0600?  20.0806? 

3.  How  many  decimal  places  are  there  in  each  of  the  prod- 
ucts of  problem  1? 

4.  Compare  the  number  of  decimal  places  in  each  of  the  prod- 
ucts of  problem  1  with  the  sum  of  the  numbers  of  decinml  places 
in  both  the  multiplicand  and  the  multiplier.     What  do  you  find? 

5.  Make  a  rule  for  finding  how  many  decimal  places  there 
must  be  in  the  product  of  two  decimals. 

6.  How,  then,  can  you  find  where  the  decimal  point  belongs 
in  the  product  of  any  two  decimals? 


DECIMAL    FRACTION'S  209 

PRINCIPLE ^11. — The  member  of  decimal  places  in  the  product 
equals  the  sum  of  the  numbers  of  decimal  places  in  the  factors. 

7.  A    field    containing    38.75  A.  yielded  23.9  bu.  of   wheat 
per  acre ;  what  was  the  total  yield? 

SOLUTION.— Since  each  acre  yielded  23.9  bu.,  38.75  A.  must  yield 
38.75  X  23.9  bushels. 

CONVENIENT  FORM          EXPLANATION. — 

38. 75  3875        339  x  38. 75  =  how  many  times  23. 9  X  38. 75? 

23.9  239         239X3875=  "  "       239x38.75? 

~TT95        "1195         239X3875=  "       23.9x38.75? 

1673  1673          What  is  i0U  of  926,125? 

1912  1912  Does  the  rule  of  problem  5  hold  true  here? 

717  717 

926.125       926125 
Ans.  926. 125  bu. 

8.  A  steer  weighing  16.22  cwt.  sold  at  $8.85'  per  cwt. ;  what 
price  did  he  bring? 

0.  A  passenger  train  ran  for  12.27  min.  at  the  rate  of  65.75 
mi.  per  minute;  how  far  did  it  run  during  the  time? 

§135.  Force  Needed  to  Draw  Loads  on  Road  Wagon. 

1.  To  draw  a  load  in  a  road  wagon  at  a  slow  walk  over  hard, 
level  country  roads  a  horizontal  pull  of  .075  of  the  total  weight 
of  the  wagon  and  load  is  required.     What  horizontal  pulls  will  be 
required  to  draw  the  following : 

WEIGHT  WEIGHT  TOTAL  HORIZONTAL 

OF  WAGON  OP  LOAD  WEIGHT  PULL  IN  LB. 

(1)  1068  lb.  2168  Ib.  

(2)  969.5  2408.3  

(3)  1580.75  3675.6 

(4)  2368  3890 

2.  Over  fresh  earth  .125    of   the  total   load  as  a   horizontal 
pull  will  draw  it  on  a  road  wagon.     Fill  out  the  vacant  columns, 
for  these  conditions : 

WEIGHT  WEIGHT  TOTAL  HORIZONTAL 

OF  WAGON  OF  LOAD  WEIGHT  PULL  IN  LB. 

(1)  980  lb.  1260  lb. 

(2)  940.8  768.78 

(3)  1164  3890.85 

(4)  1675  4060.75  ....  


210 


RATIONAL   GRAMMAR   SCHOOL   ARITHMETIC 


3.   Over  loose  sand  .258  of  the  total  load  will  draw  it. 
the  forces  needed  to  draw  these  loads : 


Find 


WEIGHT  OF 
WAGON 

WEIGHT  OF 
LOAD 

(1)    375  Ib. 
(2)  1860 
(3)  2800 

685.9  Ib. 
2586.38 
3869.25 

TOTAL 
WEIGHT 


HORIZONTAL 
PULL  IN  LB. 


4.  On  good,  broken  stone  pavement  the  pull  is  about  .0285  of 
the  total  load;  on  wood  pavement  it  is  .019  of  the  total  load; 
and  on  Macadam  pavement  it  is  .0333  of  the  load.  Find  the 
pull  in  pounds  needed  for  each  of  these  three  kinds  of  pavement 
for  the  following : 


WEIGHT  OF 
WAGON 

WEIGHT  or 
LOAD 

(1)   4060  Ib. 
(2)  5190 
(3)   4960 

3795.65  Ib. 
6340.86 
7640.65 

TOTAL 
WEIGHT 


HORIZONTAL 
PULL  IN  LB. 


§136.  Division  by  an  Integer. 

1.   I  paid  $941.25  for  15  A.  of  land;    what  was  the  price  per 
acre? 


CONVENIENT  FORM 


With  Decimals 

With  Integers 

62.75 

15)941.25 
90 

-  6275 

15)94125 
90 

41 
30 

41 
30 

its 

10.5 

112 
105 

.75 
.75 


75 
75 


EXPLANATION.— Compare  the  steps  in 
the  work  with  decimals  with  the  corre- 
sponding steps  in  the  work  [with  integers. 

How  do  the  decimal  points  stand 
through  the  problem?  How  does  the  point 
stand  in  the  quotient?  How  may  the 
dividend  be  found  from  the  divisor  and  the 
quotient?  How,  then,  may  you  check 
division? 


Ans.  $62.75. 


2.  I  paid  $2823.75  for  45  A.  of  land;  what  was  the  price  paid 


per  acre; 


DECIMAL    FRACTIONS 


211 


3.  The  36  members  of  a  society  were  assessed  equally  to  meet 
a  debt  of  $1341  against  the  society.  What  was  the  amount  of  the 
assessment  against  each  member? 


SOLUTION.  — 
37.25 


36)1341.00 
108 

261 
252 


9.0 

7.2 

llJO 
1.80 


Notice  particularly  how  by  writing  zeros  after  the  deci- 
mal point  the  quotient  may  be  carried  out  in  a  decimal 
form. 

What  effect  does  writing  zeros  after  the  point  have  on 
a  number? 


Ans,   137.25. 


4.  The  number  of  states  and  territories  and  the  total  areas  in 
square  miles  of  the  land  surface  of  the  principal  geographic  divisions 
of  continental  United  States  according  to  the  Twelfth  Census,  are 
given  in  the  following  table.  Find  the  average  land  surface  of  a 
state  for  each  division  and  for  the  whole  United  States.  Carry 
the  division  to  three  decimal  places.* 


DIVISION 

NUMBER 

LAND  SURFACE 

AVERAGE 

North  Atlantic  

9 

162,103 

South  Atlantic 

9 

168  620 

North  Central 

12 

753,550 

South  Central 

9 

610,215 

Western   

11 

1,175,742 

Continental  United  States 

5.  The  10  loads  given  in  pounds  in  column  2  of  the  following 
table  required  the  number  of  pounds  of  force  given  in  the  third 
column,  to  draw  them  on  a  common  road  wagon  over  a  good,  level 
country  road.  For  example,  1,400  Ib.  required  98  Ib.  to  pull  it; 


*If  the  next  decimal  place  after  the  last  one  required  is  less  than  5,  write  the  given 
quotient  as  the  required  decimal.  If  it  is  greater  than  5,  add  1  to  the  last  figure  of 
the  decimal  required.  This  rule  applies  to  all  cases  of  division  of  decimals. 


212  RATIONAL    GRAMMAR   SCHOOL   ARITHMETIC 

1616  lb.  required  112  lb.,  and  so  on.  Divide  each  load  by  the 
number  of  pounds  of  force  needed  to  draw  it,  and  put  the  quotient 
to  two  decimal  places  in  the  column  headed  "Katie." 

EXPERIMENT  LOAD  PULL          RATIO 

1  1400          93 

2  1616  112         

3  1825  126         

4  2236  155         

5  2440  170         .... 

6  2650  185         

7  2863  198 

8  3072  216 

9  3281  229 

10         3662         244         

§137.  Division  by  a  Decimal. 

1.  What  is  TV  of  55? 

2.  When  no  decimal  point  is  written  with  a  number  where  is 
it  understood  to  be? 

3.  Where  is  the  decimal  point  in  55? 

4.  How  is  a  number  changed  by  moving  its  decimal  point  1 
place  toward  the  left?  3  places  toward  the  left?  1  place  toward  the 
right?  2  places  toward  the  right? 

5.  55  equals  how  many  times  5.5?  .55?  .055? 

6.  If  55  -  11  =  5,  to  what  is  5.5  - 11  equal?  ,55  -  11? 

7.  Name  these  quotients:  250-25;  25-25;  2.5-25;  .25-25. 

8.  Name  these  quotients:     250  -  2.5;     25  -  2.5;     25  -  .25; 
250 -.25. 

9.  2176  -  32  =  68;  to  what  number  is  x  equal  in  each  of  the 
following  equations : 

(1)  217.6- 32  =  z;    (4)   217.6-  3.2  =  x;    (7)  2176  -    3.2  =  x\ 

(2)  21. 76- 32  =  z;    (5)  217.6  -.32  =  2;    (8)  21.76-    .32  =  x\ 

(3)  2.176-32  =  ^;    (6)  21.76-3.2  =  2;    (9)   2.176  -  .032  =  x? 

10.  In  each  case  of  problem  9  compare  the  number  of  decimal 
places  in  the  quotient  with  the  number  of  decimal  places  in  the 
dividend  minus  the  number  in  the  divisor. 


DECIMAL    FRACTIONS 


213 


11.  101,388-426  =  238;  without  dividing,  write  out  the  numbers 
to  which  x  is  equal  in  the  following  equations : 

(1)  1013S.8-426  =  z;  (4)  101388  -42.6  =  z;  (7)  101.388-  42.6  =  z; 

(2)  1013.88-426  =  ^;  (5)  101388  -4.26=£;   (8)  10.1388-   .426=z; 

(3)  1.01388-426  =  ^;  (6)  1013.88-4.26  =  ^;  (9)  1.01388-.0426  =  z. 

12.  From  the  results  of  problems  10  and  11  make  a  rule  for 
finding  the  number  of  decimal  places  in  the  quotient. 

PRINCIPLE  III. — The  number  of  decimal  places  in  the  quotient 
equals  the  number  of  decimal  places  in  the* dividend  minus  the  num- 
ber of  decimal  places  in  the  divisor. 

§138.  Problems. 

1.  In  1891  the  total  imports  of  tea  into 
U.  S.  were  82,395,924  Ib.  and  of  coffee 
511,041,459  Ib.     If  the  average  cost  of  tea 
was  37^  per  Ib.,  and  of  coffee  18^,  what 
was  the  total  cost  of  both? 

2.  On  Dec.  6,  1901,  a  carload  of  34 
Angus  show  cattle  of  weights  given  in  table 
annexed  sold  at  the  prices  per  cwt.  set 
beside  the  weights.     What  was  the  total 
weight  of  the  carload?  the  average  weight 
of  the  animals? 

3.  What  was  the  average    price    per 
hundredweight? 

4.  For  how  much  did  the  entire  load 
sell? 

5.  What   was    the  average  price  per 
head  which  the  owner  received  for  the 
carload? 

6.  The   previous  year    on  the    same 
occasion,  a  carload  of   26    prize- winning 
Angus  cattle,  averaging  1492  Ib.,  sold  at 
$15.50  per  hundredweight.     What  aver- 
age price  did  the  cattle  bring  the  owner? 

7.  How  much  did  he  receive  for  the 
carload,  problem  6? 


No. 

WEIGHT 

PRICE 

1 

1503 

$  9.00 

2 

1622 

8.85 

3 

1606 

8.60 

4 

1524 

8.50 

5 

1524 

8.50 

6 

1504 

8.10 

7 

936 

8.70 

8 

1327 

6.85 

9 

1273 

8.05 

10 

1130 

8.75 

11 

1326 

7.65 

12 

1141 

8.50 

13 

1318 

8.75 

14 

1327 

8.70 

15 

1190 

7.70 

16 

1446 

7.85 

17 

1449 

7.85 

18 

1542 

7.70 

19 

980 

6.80 

20 

1450 

8.10 

21 

852 

8.30 

22 

1376 

8.30 

23 

1540 

25.00 

24 

1631 

9.30 

25 

1468 

8.65 

26 

1297 

8.50 

27 

1529 

8.20 

28 

1100 

7.60 

29 

1073 

7.60 

30 

1110 

8.10 

31 

1095 

8.15 

32 

1456 

8.75 

33 

1298 

8.00 

34 

2130 

10.75 

214 


RATIONAL    GRAMMAR    SCHOOL    ARITHMETIC 


8.  A  dairyman  finds  that  during  November  one  of  his  cows 
furnished  this  record  : 


MORNIM;  MILKINGS 

EVENING  MILKINGS 

1st  week  

47  2  Ib 

40  4  Ib 

«2d        .    .    . 

58  6  " 

48  0  " 

3d      "       

53  8  " 

49  8  " 

4th    "     

62  7  " 

47  3  " 

29th  and  30th  days  

17.8  " 

16  4  " 

What  was  the  total  number  of  pounds  of  milk  given  by  this 
cow  during  the  month  at  the  morning  milkings?  at  the  evening 
milkings?  at  both  milkings? 

9.  If  milk  was  worth  6.25^  a  quart  (8.6  Ib.  per  gallon),  what 
was  the  milk  of  this  one  cow  worth  to  the  owner  during  November? 

§139.  Ratio  of  Circumference  of  Circle  to  Diameter. 

1.  The  distance  around  the  rung  of  a  chair  was  measured  and 
found  to  be  3.625  in.;  the  diameter  was  found  to  be  1.15.'}  in. 
Divide  the  distance  around  the  rung  by  the  diameter  and  find  the 
quotient  to  3  decimal  places. 

DEFINITION. — The  distance  around  a  circle  is  called  the  circumference 
of  the  circle. 

2.  The  circumference  of  a  circular  rod,  1.875  in.  in  diameter, 
was   measured   and   found   to   be   5.884  in.      Find  the  ratio  to 
3  decimal  places,  of  the  circumference  to  the  diameter. 

3.  Measure  the  diameters  and  the  circumferences  of  any  circles 
in  your  schoolroom  and  find  to  3  decimal  places  the  ratio  of  their 
circumferences  to  their  diameters.     If  no  circles  are  at  hand  use 
the  measures  of  this  table : 


OBJECT 

I'IRCITMFERENCE 

DIAMETER 

RATIO 

Ink  bottle         .... 

5  5      in 

1  75    in 

Tin  box           

r>  ir,.j 

1  1)5:; 

Globe  of  lamp  
Terrestrial  globe  
Barrel  -head        .  .  . 

28.588 

57.080 
71  458 

8.444 
18.0f>',} 
22  750 

Iron  rincj 

18  125 

5  675 

Avorace 

DECIMAL    FRACTIONS 


215 


4.  Measure  the  circumferences  and  the  diameters  of  the  fol- 
lowing objects  and  find  the  ratios  to  three  decimal  places.  If  you 
are  unable  to  make  your  own  measures  use  those  of  the  table: 


OBJECT 

ClBCUMFEKENCE 

IMAMKTKK 

RATIO 

I»i<'V('le  wh66l 

81  177 

26  025 

Bicycle  wheel  

88.055 

28.012 

Front  carriage  wheel 
Rear 
Locomotive  driver.  . 
Average 

151.189 
176.333 
174.762 

48.125 
56.125 
55.625 

5.  Find  the  ratio  of  the  circumferences  to  the  diameters  of  the 
coins  of  problem  27,  p.  114. 

G.  It  is  proved  in  Geometry  that  the  ratio  of  the  circumference 
of  any  circle  to  its  diameter  is  about  3^,  or  more  accurately  3.1416. 
If  then,  the  diameter  of  a  circle  is  known,  how  may  the  circum- 
ference be  found  without  measurement? 

7.  Find  the  average  of  all  the  values  of  the  ratios  found  in  prob- 
lems 1  to  4  and  compare  this  last  average  with  3|;  with  3.1416. 
What  is  the  difference  in  each  case? 

NOTE. — For  most  practical  purposes  this  ratio,  denoted  by  the  Greek 
letter  TT,  and  called  pi,  may  be  taken  as  3).  For  greater  accuracy  use  ir  = 
3.1416. 

§140.  Original  Problems. 

Make  and  solve  problems  based  on  the  following  facts: 

1.  1  cu.  ft.  is  about  .8  of  a  bushel  of  small  grain.     A  grain  bin 
is  8'  x  12'  x  22'. 

2.  1  bu.  of  ear  corn  is  about  2.25  cu.  ft.    A  wagon  box  is  2. 67' 
x  2.85' x  9.6'. 

3.  Well  settled  timothy  hay  runs  about  355.25  cu.  ft.  to  the  ton. 
A  hay  shed  is  24'  x  40'  and  is  filled  with  hay  to  a  height  of  18.5'. 

4.  Loose  timothy  hay  runs  about  460  cu.  ft.   to  the  ton.     A 
load  of  hay  is  8'  x  18'  x  22.7'. 

5.  Stove  coal  runs  about  35.1  cu.  ft.  to  the  ton.     A  coal  bin  is 
6' x  12.5' x  15. 35'. 

6.  1  perch  of  stone  is  24.75  cu.  ft.     A  stone  wall  is  2.75'x 
4.385'  x  126.8'. 


216 


RATIONAL    GRAMMAR    SCHOOL    ARITHMETIC 


7.  A  man  walks  about  3.5  mi.  per  hour.     It  is  85  mi.  from 
Chicago  to  Milwaukee. 

8.  A  horse  trots  about  7.5  mi.  per  hour. 

9.  A  horse  runs  about  18  mi.  per  hour  for  short  distances. 

10.  A  steamboat  runs  18  mi.  per  hour. 

11.  A  slow  river  flows  3  mi.  per  hour. 

12.  A  rapid  river  flows  7  mi.  per  hour. 

13.  A   crow  flies  25  mi.  per  hour;  a  falcon,  75  mi.;   a  wild 
duck,  90  mi. ;  a  sparrow,  92  mi. ;   and  a  hawk,  150  mi.  per  hour. 

14.  A  carrier  pigeon  flies  80  mi.  per  hour  for  long  distances. 

15.  Sound  travels  through  air  1134  ft.   per  second;  through 
water,  5000  ft.  per  second;   and  through  iron  or  steel,  17,000  ft. 
per  second. 

16.  A  rifle  ball  travels  1460  ft.  per  second  at  starting;  and  a 
20-lb.  cannon  ball,  16,000  ft.  per  second  at  starting. 

17.  Light  travels  186,600  mi.  per  second;  electricity,  288,000 
mi.  per  second.     The  sun  is  93,000,000  mi.  from  the  earth;   the 
moon,  240,000  miles. 

18.  One  horse-power  raises  33,000  Ib.  through  a  height  of  1  ft. 
in  1  minute. 

19.  The  equatorial  diameter  of  the  earth  is  7925.6  mi.;   the 
polar  diameter,  7899.1  miles. 

§141.  Physical  Measurements. 

The  following  table  gives,  for  men,  of  ages  from  18  to  26  years, 
the  average  lung  capacity  in  cubic  inches,  the  height  in  inches, 
and  the  weight  in  pounds. 


AGE 

LUNG  CAPACITY 

HEIGHT 

WEIGHT 

18 

251.4 

68.2 

134.25 

19 

251.8 

68.2 

135.40 

20 

258.2 

68.1 

138.65 

21 

260.4 

68.1 

140.60 

22 

264.8 

68.2 

141.15 

23 

263.7 

68.1 

138.60 

24 

267.1 

68.2 

143.90 

25 

267.2 

68.2 

143.15 

26 

267.1 

68.9 

142.30 

DECIMAL    FRACTIONS 


217 


1.  Find  the  number  of  cubic  inches  of  lung  capacity  per  inch 
of  height,  for  each   age,  by  computing  the  ratio  to  2  decimal 
places  of  each  number  of  column  2  to  the  corresponding  number  of 
column  3.     Is  this  ratio  the. same  for  all  ages? 

2.  Similarly  find  the  number  of  cubic  inches,  to  2  decimal 
places,  of  lung  capacity  per  pound  of  weight,  for  each  age. 

3.  For  each  age  find  the  number  of  pounds  of  weight  per 
inch  of  height  to  2  decimal  places.     Are  these  numbers  the  same 
for  all  ages? 

§142.  Specific  Gravity. 

The  specific  gravity  of  any  solid  or  liquid  substance  is  the 
ratio  of  its  weight  to  the  weight  of  an  equal  bulk  of  water.  The 
weight  of  a  cubic  foot  of  water  is  62.5  pounds. 

1.  The  following  table  contains  the  weight  in  pounds  of  1  cu. 
ft.  of  the  substances  mentioned.  Find  to  3  decimal  places  the 
specific  gravities  of  these  substances : 


METAL, 

WEIGHT  OF 
1  Cu.  FT. 

SPECIFIC 
GRAVITY 

WOOD 

WEIGHT  OF 
1  Cu.  FT. 

SPECIFIC 
GRAVITY 

LIQUID 

WEIGHT  or 
1  Cu.  FT. 

SPECIFIC 
GRAVITY 

Aluminum  . 

166  5 

Cork  

15.00 

Alcohol  

50  0 

Zinc      

436  5 

Spruce 

31  25 

Turpentine 

54  4 

Cast  iron  .  .  . 
Tin  

450.0 
458  3 

Pine  (yellow) 
Cedar 

34.60 
35  06 

Petroleum  .    . 
Olive  oil 

55.7 

57  0 

Wr't  iron.  .  . 
Steel   

480.0 
490  0 

Pine  (white) 
Walnut  .  . 

28.00 
41  90 

Linseed  oil  .    . 
Sea  water  .  .   . 

59.5 
64  1 

Brass  

523  8 

Maple  .   ..   . 

46  88 

Milk  

64  5 

Copper  

552.0 

Ash  

52  80 

Acetic  acid  .  . 

66.5 

Silver    . 

655  1 

Beech 

53  25 

Muriatic  acid 

75  0 

Lead   

709  4 

Oak   

65  00 

Nitric  acid 

95  0 

Gold      .    . 

1200  9 

Ebony 

76  00 

Sulphuric  acid 

115  1 

Platinum  .  . 

1347.0 

Lignum  vitse 

83.30 

Mercury  

880.0 

2.  The  specific  gravity  of  any  gas  or  vapor,  is  the  ratio  of  its 
weight  to  the  weight  of  an  equal  volume  of  air,  the  gas  or  vapor 
and  the  air  being  at  the  same  temperature  and  under  the  same 


RATIONAL    BKAMMAR    SCHOOL    ABI1HMETIO 


pressure.     Find  to  3  decimal  places  the  specific  gravities  of  the 
gases  and  vapors  in  the  following  table: 


GAS 

WEIGHT,  IN  POUNDS, 
OF  l  CU-BIC  FOOT 

SPKOIFIC  UKAVITV 

Hydrogen 

00559 

Srnoke  (wood) 

00727 

Smoke  (soft  coal)  

.00815 

Steam  at  212    F 

03790 

Carbonic  oxide         

07810 

Nitrogen 

07860 

Air  .    .           .... 

08073 

Oxygen  .    .                .    . 

08925 

Carbonic  acid.   . 

12344 

Chlorine  

19700 

3.  The  following  familiar  substances  may  be  compared  with 
water  as  to  weight.  Find  their  specific  gravities  to  2  decimal 
places : 


SUBSTANCE 

WEIGHT,  IN  POUNDS, 
OF  1  CUBIC  FOOT 

SPECIFIC  GRAVITY 

Glass  (average) 

175  8 

Chalk  

174  5 

Marble  .  .         .       

169  2 

Granite  

166  4 

Stone  (common)  

158  2 

Salt             " 

133  4 

Soil             "           

124  5 

Clay  

121  8 

Brick  

118  3 

Sand  

113  9 

§143.  To  Eeduce  a  Common  Fraction  to  a  Decimal. 

1.  Express  $f  decimally. 

SOLUTION 
.375 


8)3.000 
2.4 

760 
.56 


EXPLANATION.  —  Annex  zeros  to  the  right  of  the 
decimal  point  after  the  numerator,  and  then  divide  by 
the  denominator. 


Ans. 


1.375 


.040 
.040 


DECIMAL   FRACTIONS  211) 

2.  Express   the  following  common  fractions  decimally  (to  3 
decimal  places) : 

f.     2  .     4  .     7  .     5  .      3    .      5.13 
)    3>    ~5J    8>    8">    TT>    ~5Z>    64* 

3.  Express   the   following    mixed    numbers   decimally    (to   3 
places) : 

H;  3J;  2i;  6f;  13^;  18Jf 

4.  Express  the  following  decimally  to  5  decimal  places : 

i;  A;  A;  A;  H;  A- 

Such  decimals  as  these  fractions  give  rise  to,  that  do  not  ter- 
minate, are  called  non-terminating  decimals. 

5.  Express   the   following   numbers   decimally   to    6    decimal 
places : 

i;  !;  l-;i;  i;  I;  i;  A;  A;  A;  iS;  it;  f*. 

DEFINITION.— Such  non-terminating  decimals  as  these  that  repeat  the 
same  digit  or  group  of  digits  indefinitely,  are  called  repetends,  or  circula- 
ting decimals,  or  circulates. 

6.  Express  the  values  of  these  numbers  by  the  use  of  integers 
and  decimals  only : 

1.3$;  10.87*;  5.28| ;  17.UJ;  20.0f;  l.UOJ;  -OOJ;  .0-J;  .000^; 
30.060|. 

7.  Express  the  following  fractions  decimally  to  4  places : 

11.     I-       7.3.     42.     3*.    M.        8'65.        ^ 

'  7*'  7.8'  13.17'  12i' 

8.  A  man   sold  37^  A.   of  his  farm  of  79f  A. ;    how  many 
hundredths  of  his  farm  did  he  sell? 

9.  A  man  owned  a  piece  of  city  land  450'  square.     A  strip 
37J-'  wide  was  cut  from  each  of  its  four  sides  for  streets.     How- 
many  hundredths  of  the  square  were  cut  away? 

10.  How  many  hundredths  of  the  area  of  a  page  of  this  book 
are  in  the  margins?     (Measure  to  the  nearest  16th  of  an  inch). 

11.  How  many  hundredths  of  the  area  of  the  surface  of  your 
desk  is  the  area  of  the  surface  of  your  book? 

12.  How  many  hundredths  of  the  area  of  the  surface  of  the 
floor  is  the  area  of  the  surface  of  your  desk? 


KATlUJSALi    lilt  AMM  Alt    SUHUULi 


§144.  Area  of  a  Circle. 

Draw   a   circle  and   diameter  AB  (see  Fig.   132).     With  an 
opening  of  the  compasses  equal  to  the  radius  of  the  circle  and  with 


^ 


FIGURE  132 

A  as  center  draw  short  arcs  at  D  and  6r,  also,  with  same  radius 
and  with  B  as  center,  draw  short  arcs  at  E  and  F.  Draw 
radii  OD,  OE,  OF,  and  OG. 

Bisect  angle  A  OD,  as  in  Problem  II,  p.  186,  and  draw  OS. 

With  an  opening  of  the  compasses  equal  to  the  distance 
between  D  and  H  and  with  center  D  draw  an  arc  at  «7;  with 
center  B  draw  arcs  at  A" and  L\  also,  with  center  G  draw  arcs  at 
M  and  N.  Draw  radii  O/,  OK,  OL,  OM,  and  ON. 

Cut  the  circle  into  the  twelve  equal  sectors  thus  formed  and 
place  these  sectors  as  shown  in  the  second  part  of  the  figure. 
How  long  is  the  base  AB  of  the  approximate  parallelogram  thus 
formed?  How  wide  is  the  approximate  parallelogram? 

If  each  of  the  twelve  sectors  were  split  into  halves  and  the 
resulting  twenty -four  sectors  were  fitted  together  as  are  the  twelve 
sectors,  would  the  wavy  base  line  become  more  nearly  straight? 
How  long  and  how  wide  would  the  new  approximate  parallelogram 
be?  What  would  be  its  area? 

Having  the  circumference  and  the  radius  of  a  circle  how  can 
you  find  the  area  of  the  circle? 

Having  the  radius  of  any  circle  how  can  you  find  the  area  of 
the  circle  (See  §138)? 

PROBLEMS 

In  problems  1  to  3,  inclusive,  use  tr  =  3|.  Let  r  denote  the 
length  of  the  radius  of  a  circle,  let  c  denote  its  circumference, 
and  A,  its  area. 

1.  Find  the  areas,  A,  of  these  circles 

(1)  r  =  12.5',     c  =  78.54';     (3)  r  -  20",  c  =  125.664"; 

(2)  r  =    6.25',  c  -  39.27';     (4)  r  =  60",  c  =  377.143". 


DECIMAL   FRACTIONS  221 

2.  The  diameter  of  a  drum-head  is  2.5'  and  the  circumference 
is  7.854',  how  many  square  inches  of  skin  are  in  the  two  heads? 

3.  The  diameter  of  a  circular  window  is  18.5"  and  its  circum- 
ference is  58.12",  what  is  its  area? 

In  the  following  problems  TT   is   taken  as  3.1416.      Results 
should  be  correct  to  4  decimal  places. 

4.  The  wind  is  blowing  squarely  against  a  circular  signboard, 
whose  diameter  is   32.75  ft.,  with  a  pressure  of  25.5  Ib.  to  the 
square  foot.     Find  the  total  wind  pressure  against  the  board. 

5.  The  circumference  of  a  cylindrical  chair  rung  is  5.5";  find 
the  diameter  and  area  of  the  right  section  of  the  rung. 

NOTE. — A  right  section  is  the  section  that  would  be  made  by  sawing 
the  rung  square  across. 

6.  Steam  passes  from  the  boiler  of  an  engine  through   the 

passage,  P,  into  the  cylinder,  and  pushes 
against  the  circular  piston,  (7,  with  a  pres- 
sure of  65  Ib.  to  the  square  inch.  If  the 
diameter  of  the  circular  head,  (?,  is  12.5", 
what  is  the  total  pressure  against  the  left  side 

FIGURE  133  of    the  pist(m? 

7.  Answer   similar  questions  for  pistons  having  these  diam- 
eters with  steam  pressures  per  square  inch  as  indicated: 

DIAMETER  OF  PRESSURE  PER 

PISTON  SQUARE  INCH 

(1)  10.85"  86 

(2)  22.35"  120 

(3)  14.85"  180 

(4)  16.45"  175 

(5)  20.75"  125 

(6)  15.85"  160 

(7)  21.35"  205 

8.  Find  the  area,  ^4,  of  a  circle  whose  radius  is  r  ft.  long  and 
whose  circumference  is  c  ft.  long. 

9.  Find  the  circumference,    c,    of  a  circle  whose  radius  is 
r  rods  long.     Find  the  area. 

10.  A  cow  is  tied  to  the  corner  of  a  corn  crib,  18'  x  18',  with 
a  rope  18'  long.     Over  how  many  square  feet  can  she  graze? 


RATIONAL   GRAMMAR   SCHOOL  ARITHMETIC 

§145.  Gear  of  Bicycle. 

In  this  section  use  ?r  =  s-f-. 

1.  How  many  times  does  the  large  wheel  of  a  high  bicycle 
revolve  while  the  cranks  turn  round  once? 


Aa 
FIGURE  134 

2.  If  the  diameter  of  the  driving  wheel   is  5.5  ft.,  how  far 
will  the  high  bicycle  advance  while  the  cranks  turn  round  once? 

3.  If  there  are  7  teeth  on  the  rear  sprocket  of  a  safety  bicycle 
and  14  teeth  on  the  front  sprocket,  how  many  times  will  the  driv- 
ing wheel  revolve  while  the  cranks  turn  round  once? 

4.  The  diameter  of  the  driving  wheel  of  a  safety  bicycle  is 
28  inches.    If  there  are  21  teeth  on  the  front  sprocket  and  7  on 
the  rear,  how  far  forward  will  one  complete  turn  of  the  cranks 
carry  the  bicycle?     How  far  if  there  are  28  and  8  teeth? 

5.  Answer  similar  questions  if  the  front  and  the  rear  sprockets 
have  24  and  9  teeth  respectively;  30  and  9;  33  and  9. 

6.  Answer  similar  questions  if  the  length  of  the  tire  of  the 
driving  wheel  is  6.85',  sprockets  having  teeth  as  in  problem  5. 

7.  How  long  is  the  diameter  of  a  wheel  if  one  revolution  car- 
ries it  along  24.09  ft.  on  the  ground? 

8.  I  roll  my  bicycle  along  until  the  driving  wheel  turns  over 
just  once.    The  distance  traversed  is  7'  4".    What  is  its  diameter? 

9.  How  long  is  the  tire  of  the  driving  wheel  of  a  high  bicycle 
(see  Fig.  134)  if  its  diameter  is  56"?  08"?  72"?  80"?  45"? 

10.  How  far  is  a  high  bicycle  moved  forward  by  1  turn  of  its 
84"  driving  wheel? 

NOTE — A  28"-bicycle  means  a  bicycle  whose  driving  wheel  is  28"  in 
diameter. 

11.  If  there   are   29   teeth   on  the  front  and  7  on  the  real- 
sprocket  of  a  28"-bicycle  how  far  will  one  turn  carry  it? 


DECIMAL   FRACTIONS 


223 


A  safety  bicycle  is  said  to  be  "geared"  to  72",  84"  and  so  on 
when  one  turn  of  the  cranks  would  carry  it  just  as  far  forward  as 
would  one  turn  of  a  wheel  having  a  diameter  of  72",  84"  and 
so  on. 

12.  Count  the  teeth  on  the  front  and  the  rear  sprockets  of 
some  safety  bicycle,  measure  the  diameter  of  the  driving  wheel, 
and  find  its  "gear." 

13.  Find  to  2  decimal  places  the  "gear"  of  these  safety  bicycles: 


NUMBER  or  TEETH 

DIAMETEK 
DRIV.  WHEEL 

GEAR 

Rear  Sprocket 

Front  Sprocket, 

18' 

7 

14 

22' 

7 

24 

24' 

7 

26 

26' 

8 

28 

26' 

9 

32 

28' 

7 

28 

28' 

8 

28 

28' 

8 

34 

§146.  Law  Formulated— (ALGEBRA). 

1.  If  the  diameter  of   the  driving  wheel  of  a  safety  bicycle 
is   d,  and   the   ratio  of   the  circumference  to  the  diameter  of  a 
circle  is  TT,  write  an  equation  showing  the  length,  c,  of  the  cir- 
cumference. 

2.  If  t  is  the  number  of  teeth  on  the  rear  sprocket,  and  J'the 
number  on  the  front  sprocket,  write  an  equation   showing  the 
number,  -w,  of  times  the  driving  wheel  will  turn  while  the  cranks 
turn  once. 

3.  Write  an  equation  showing  the  distance,  d,  that  one  revolu- 
tion of  the  cranks  will  carry  the  bicycle  forward,  using  the  letters 
of  problems  1  and  2. 

4.  Calling  G  the  gear  of  a  safety  bicycle,  show  that  the  gear  is 

T 

given  by  the  equation  G  =  —  x  d,   the  letters   meaning  the  same 

as  above. 

5.  Make  a  rule  for  finding  the  gear  of  a  safety  bicycle  the 
diameter  of  whose  driving  wheel  is  d  feet. 


224 


RATIONAL   GRAMMAR   SCHOOL    ARITHMETIC 


§147.  Comparison  of  Sail  Areas  of  Yachts. 

The  sail  areas  of  the  four  racing  yachts,  Genesee,  Yankee, 
Illinois,  and  Milwaukee,  were  computed  from  the  measurements 
indicated  in  the  drawing  (Fig.  135). 


L.w.L.aa' 

Yankee  Illinois 

FIGURE  135 


L.W.L  37/84 

Milwaukee 


1.  The  small  sails  in  front  are  called  jibs.     The  base  and  the 
altitude  of  the  jibs  of  each  yacht  are  indicated  in  Fig.  135.     Find 
the  number  of  square  feet  in  the  jib  for  each  of  the  four  yachts. 

2.  Genesee  was   cup-winner.     Find   the   ratio,  to  1    decimal 
place,  of  the  sail  area  of  the  jib  of  Yankee  to  that  of  Genesee. 

3.  Find  a  similar  ratio  for  each  of  the  other  two  yachts. 

4.  The  jibs  were  made  of  a  kind  of  coarse  cloth,  called  duck, 
which  is  sold  in  strips  2  ft.  wide  and  weighs  8  oz.  per  linear 
yard  (per  yard  of  length).     Find  the  weight  of  1  sq.  ft.  and  the 
weight  of  the  jib  of  each  of  the  four  yachts. 

5.  To  compute  the  areas  of  the  main  sails,  each  was  divided  into 
two  triangles,  like  AEB  and  ABC  of  Genesee;  the  bases  and  the 
altitudes  were  measured  and  found   as  indicated  in  the  figures. 
The  altitudes,  AF  and  AD,  and  bases,  EB  and  BC,  of  Genesee 
having  the  indicated  lengths,  find  the  total  area  of  her  mainsail. 

6.  Similarly  find  the  total    area  of   the   mainsail  of  each  of 
the  other  three  yachts. 

7.  These  mainsails  weigh  10  oz.  per  linear  yard  (strips  2  ft. 
wide).     Find  the  weight  of  the  mainsail  of  each  yacht. 

8.  Find  to  2  decimals  the  ratio  of  the  area  of  the  mainsail  of 
the  Yankee  to  that  of  the  Genesee  (the  winner). 

9.  Similarly,  find  the  ratio  of  the  mainsail  area  of  each  of 
the  other  two  yachts  to  the  mainsail  area  of  Genesee. 

10.  Find  the  total  sail  areas  and  sail  weights  of  each  yacht. 


COMPOUND    DENOMINATE    NUMBERS  225 

COMPOUND  DENOMINATE  NUMBERS 

§148.  Definitions. 

A  denominate  number  is  a  number  whose  unit  is  concrete; 
as,  13  mi.,  8  hr.,  40  A.,  $125,  etc. 

A  concrete  unit  is  a  unit  having  a  specific  name;  as,  1  yd., 
1  lb.,  $1,  1  hat,  1  horse,  etc. 

A  compound  denominate  number  is  a  number  expressed  in  two  or 
more  units  of  the  same  kind;  as,  11  hr.  25  min.  15  sec.;  12  gal. 
3  qt.  1  pt.  3  gills. 

TABLES  OF  MEASURES 

NOTE. — Read  carefully  the  tables  that  follow  and  fix  them  in  mind  by 
solving  the  problems  beginning  on  page  231. 

§149.  Measures  of  Value. 

The  standards  of  value  of  the  United  States  and  of  some  of 
the  European  countries  are  here  given  with  both  their  rough  and 
their  accurate  equivalents  in  U.  S.  money. 

ROUGH         ACCURATE 
COUNTRY  STANDARD  SYMBOL  EQUIVALENT  EQUIVALENT 

United  States  Dollar  $  $1.  $1. 

Great  Britain  Pound  (Sterling)  £  $5.00  $4.8665 

Germany  Mark  (Reichsmark)  M.  $  .25  $  .2385 

France  Franc  F.  $  .20  $  .193 

Russia  Ruble  R.  $  .75  $  .772 

Austria-Hungary    Crown,  or  Filler  C.  or  F.     $  .20  $  .203 

Italy  Lira      „  L.  $  .20  $  .193 

TABLE  OF  U.  S.  MONEY 
10  mills  (in. )  =  1  cent  (ct.  or  ^) 
10  cents          =  1  dime  (d.) 
10  dimes         =  1  dollar 
10  dollars       =  1  eagle 

5  dollars       =  ^  eagle 

2 1  dollars      =  £  eagle 
20  dollars       =  1  double  eagle 

The  coins  of  the  United  States  are  bronae,  nickel,  silver,  and 
gold. 

TABLE  OF  ENGLISH  MONEY 
4  farthings  (far.)  =  1  penny  (d.) 
12  pence  =  1  shilling  (s.) 

20  shillings  =  1  pound  (£) 

21  shillings  =  1  guinea 


226  RATIONAL  GRAMMAR  SCHOOL  ARITHMETIC 

TABLE  OF  MONETARY  UNITS  OF  OTHER  NATIONS 
Germany,  1  mark  (M.)  =100  pfennige  (pf.) 

France,  1  franc  (fr.)  =  100  centimes  (c.) 

Russia,  1  ruble  (r.)  =  100  copecks  (c.) 

Austria-Hungary,  1  crown,  or  filler  =  100  heller 
Italy,  1  lira  =100  centessimi. 

§150.  Measures  of  Weight. 

Three  systems  of  weight  units  are  used  in  the  United  States, 
viz. :  troy  weight,  avoirdupois  weight,  and  apothecaries'  weight. 

Troy  weight  is  used  in  weighing  gold,  silver,  and  jewels. 
Avoirdupois  weight  is  used  for  weighing  all  ordinary  articles,  and 
apothecaries'  weight  is  used  by  druggists  in  mixing  medicines. 

The  standard  of  weight  in  the  U.  S.  is  the  troy  pound.  The 
grain  is  the  same  in  all  three  systems;  the  pound  the  same  in  troy 
and  in  apothecaries'  weight  and  different  in  avoirdupois. 

TABLE  OF  TROY  WEIGHT  TABLE  OF  EQUIVALENTS 

24  grains  (gr.)       =  1  pennyweight  (dwt.)    5760  gr.       \ 
20  penny  weights  =  1  ounce  (oz.)  240  dwt.    /•  =  1  Ib. 

12  ounces  =  1  pound  (Ib.)  12  oz.       ) 

TABLE  OF  AVOIRDUPOIS  WEIGHT 
7000  grains  (gr.)  =  1  pound  (Ib.) 
16  ounces  =  1  pound  (Ib.) 

100  pounds  =  1  hundredweight  (cwt.) 

2000  pounds  =  1  ton  (T.) 

2240  pounds  =  1  long  ton  (L.T.) 

TABLE  OF  APOTHECARIES'  WEIGHT 
20  grains  (gr.)  =  1  scruple  (3) 

3  scruples        =  1  dram  (3) 

8  drams  =  1  ounce  (1 ) 

12  ounces          =  1  pound  (Ib.) 

§151.  Measures  of  Length,  or  Distance  (Linear  Measure). 

The  standard  unit  for  measures  of  length,  or  distance,  is  the 
yard. 

TABLE  OF  COMMON  LINEAR  MEASURE  TABLE  OF  EQUIVALENTS 

12    inches  (in.)          =  1  foot  (ft.)  63360  in. 


3    feet  =1  yard  (yd.)  5280ft.      , 

5$  yards,  or  16£  ft.  =  1  rod  (rd.)  1760  yd.     C  ~~ 

320    rods  =  1  mile  (mi.)  320  rd. 


COMPOUND    DENOMINATE    NUMBERS  227 

TABLE  OP  SURVEYORS'  LINEAR  MEASURE 

7. 92  inches  =  1  link  (li.) 

100  links         —  1  chain  (ch.) 

80  chains       =  1  mile  (mi.) 

§152.  Measures  of  Surface. 

The  unit  upon  which  surface  measure  is  based  is  the  square 
yard,  which  is  a  square  each  of  whose  sides  equals  1  yard. 

TABLE  OF  COMMON  SURFACE  MEASURE 

144    square  inches  (sq.  in.)  =  1  square  foot  (sq.  ft.) 
9    square  feet  =  1  square  yard  (sq.  yd. ) 

30£  square  yards  =  1  square  rod  (sq.  rd.) 

160    square  rods  =  1  acre  (A.) 

TABLE  OF  SURVEYORS'  SURFACE  MEASURE 

025  square  links  (sq.  li.)  =  1  square  rod  (sq.  rd.) 
Ki  square  rods  =  1  square  chain  (sq.  ch.) 

10  square  chains  =  1  acre  (A.) 

640  acres  (a  section)          =  1  square  mile  (sq.  mi.) 
36  square  miles  =  1  township  (Tp. ) 

TABLE  OF  EQUIVALENTS 
3686400  sq.  rd.     \ 

23040  sq.  ch.     I  =  1  sq.  mi. 
36  sq.  mi.    ) 

§153.  Measures  of  Volume. 

TABLE  OF  CUBIC  MEASURE 

1728  cubic  inches  (cu.  in.)  =  1  cubic  foot  (cu.  ft.) 
27  cubic  feet  =  1  cubic  yard  (cu.  yd.) 

TABLE  OF  EQUIVALENTS 
46656  cu.  in. 


,    =  1  cu. 
27  cu.  ft.    ) 

A  perch  of  stone  is  a  square-cornered  mass,  1'  x  !£'  x  16-J', 
or  24f  cubic  feet. 

Fire  wood  is  measured  by  the  cord.  A  cord  of  wood  is  a 
straight  pile,  4'  x  4'  x  8',  or  128  cubic  feet.  A  cord  foot  is  a  straight 
pile  of  wood,  4'  x  4'  x  1'.  How  many  cubic  feet  are  there  in  a 
cord  foot? 


228 


RATIONAL   GEAMMAR    SCHOOL    ARITHMETIC 


§154.  Measures  of  Capacity. 

Liquid  measure  is   used  in  measuring  liquids,  the  capacity  of 
cisterns,  tanks,  etc. 


TABLE  OF  LIQUID  MEASURE 

4  gills  (gi.)  =  l  pint  (pt.) 
2  pints          =  1  quart  (qt. ) 
4  quarts       =  1  gallon  (gal. ) 


TABLE  OP  EQUIVALENTS 
32  gi. 

8  pt.     J-  =  1  gal. 
4qt. 


NOTE. — A  liquid  gallon  contains  231  cubic  inches. 

Dry  measure  is  used  in  measuring  grain,  fruit,  vegetables,  etc. 
The  standard  of  dry  measure  is  the  Winchester  bushel,  which  is 
a  cylinder  18^  in.  in  diameter  and  8  in.  deep,  containing  2150.42 
cubic  inches.  , 


TABLE  OF  EQUIVALENTS 
64  pt.      } 
32  qt.      [  =  1  bu. 
4pk.     ) 

NOTE. — A  dry  gallon  =  4  qt.  or  \  pk.,  or  268.8  cubic  inches. 

AVOIRDUPOIS   POUNDS  IN  A  BUSHEL   PRESCRIBED  BY  VARIOUS   STATES 


TABLE  OF  DRY  MEASURE 
2  pints  (pt. )  =  1  quart  (qt.) 
8  quarts  =  1  peck  (pk. ) 
4  pecks  =  1  bushel  (bu. ) 


COMMODITIES. 


Barley 

Beans 

Blue  Grass  Seed  . 

Buckwheat 

Castor  Beans 

Clover  Seed 

Coal  (Anthracite) 
Corn  on  the  Cob  . 

Corn,  Shelled 

Cornmeal 

Dried  Apples 

Dried  Peaches  — 

Flax  Seed  

Hqmp  Seed 

Millet 

Oats 

Onions 

Peas 

Potatoes  

Rye 

Sweet  Potatoes  . . 

Timothy  Seed 

Turnips  

Wheat  . . 


50  48  48  48  48  48  47  48  48  48  48  48  48 


60  60  60  60  60  60  60  62 


14 

40  48  52 
46 


60  60  60  60  60  60 


525656 
48 
21 


80 


504850 


50 

45 

5055 


60606060 


50525056 

46  46  46 


70  68  70  70  70 


33  33  33  33  39 
36..  565656 
44  44  44  44!44 


46  50  55 
45  45  45  45  45 
5560 


80  80  76 


5050 
25  24  24  24 


48  50  50 


45 


i 

«o 
«c 

I  ^ 


48  48  48  50  52 


60 


606060 


505050 


56 


54  56  56  56  56  56  56  56  50  56  56  56  56  56  56  56  56  56  56  56  56 


5456 
45 

58 


222824 
282833 


60  60  60  62  60 


32  32  32  32  32  32  32  32  32  32  32  32  32  32  30  32  33  30  32  32  32  32  32  32 
. .  50  57  48  57  57  57  . .  52  52  5.4  . .  57  . .  57  . .  55  . .  5657  52  57|50  57 
. .  60  . .  ..  60  60  60  ..  60  ....  60  60  60  60  ..  60  ..  60  60160  . . 


60  60  60  60  60  60  60  60  60  60  60  60  60  60  60  60  60  60  60  60  60  60  60 


45 


56  56  56  56  56  56  56  56  56  56  56  56  58  56  56  56  56  56  56  56  5S 


48  48  48  47  48  48  48  48  48  48 


50  48  50  48 


64  60  60  62 


55  55  56 


50 
4445 


60  60  60  60  60 

14 
42  48  52  42  48 


7070 


50.. 

2428 

..28 

5656 

44 


5866 
50'55  .. 
45  45  45  45J40 
605550 


I! 


16060606060 
70 


70 


56 


50 

282825 
322828 


56 


COMPOUND    DENOMINATE    NUMBERS  229 

§155.  Measures  of  Time. 

The  standard  unit  for  measuring  time  is  the  mean  solar  day. 
The  mean  solar  day  is  the  average  time  interval  from  the  instant 
when  the  sun  crosses  the  meridian  of  a  place  (noon)  to  the  next 
instant  of  crossing  the  meridian  (the  next  noon). 

TABLE  OF  TIME  MEASURE 
60  seconds  (sec.)        =  1  minute  (min.) 
60  minutes  =  1  hour  (hr. ) 

24  hours  =  1  day  (da. ) 

7  days  =  1  week  (wk.) 

30  days  =  1  calendar  month  (mo.) 

12  calendar  months  =  1  calendar  year  (yr.) 

365  days  =  1  common  year 

366  days  =  1  leap  year 
100  years  =  1  century 

NOTE. — One  mean  solar  year  =  365  da.  5  hr.  48  min.  46  sec.  =  365}  da. 
(nearly). 

TABLE  OF  EQUIVALENTS 

31536000  sec.  1 

525600  min. 
8760  hr. 
365  da. 
52  wk.  1  da. 
12  mo. 

According  to  the  Julian  calendar,  adopted  by  Julius  Caesar 
(whence  the  name),  every  year  whose  date  number  (as  1896,  1904) 
is  divisible  by  4  must  contain  366  da.  and  all  other  years  contain 
365  da.  Years  containing  366  da.  are  called  leap  years;  those 
containing  365  da.  are  called  common  years. 

According  to  the  Gregorian  calendar,  which  is  now  used  by 
nearly  all  civilized  nations,  every  year  whose  date  number  is 
divisible  by  4  is  a  leap  year,  unless  the  date  number  ends  in 
two  zeros  (as  1600,  1900),  in  which  case  the  date  number  must 
be  divisible  by  400  to  be  a  leap  year. 

The  extra  day  of  the  leap  year  is  added  to  February,  giving 
this  month  29  da.  in  leap  years. 

§156.  Measurement  by  Counting. 

TABLE  FOR  COUNTING  TABLE  FOR  MEASURING  PAPER 
12  things  =  1  dozen  (doz.)  24  sheets     =  1  quire 

20  things  =  1  score  20  quires    =  1  ream 

12  dozen    =  1  gross  2  reams     =  1  bundle 

12  gross     =  1  great  gross  5  bundles  =  1  bale 


230  RATIONAL    GRAMMAR   SCHOOL    ARITHMETIC 

The  operations   to  be   performed   in   compound  denominate 
numbers  are  the  following: 

(1)  To  change  from  higher  to  lower  units,  or  denominations; 

(2)  To  change  from  lower  to  higher  units,  or  denominations; 

(3)  To  add  compound  denominate  numbers; 

(4)  To  subtract  such  numbers; 

(5)  To  multiply  such  numbers; 

(6)  To  divide  such  numbers. 

These  processes  will  now  be  illustrated  in  order. 
1.  Express  16  bu.  3  pk.  3  qt.  1  pt.  as  pints. 
CONVENIENT  FORM 

16  bu.  3  pk.  3  qt.  1  pt. 
4          =  No.  pk.  in  1  bu. 


64  pk.  =  No.  pk.  in  16  bu. 
3  pk. 


07  pk.  =  No.  pk.  in  16  bu.  7  pk. 
8          =  No.  qt.  in  1  pk. 


536  qt.    =  No.  qt.  in  67  pkv 
3qt, 

539  qt.    =  No.  qt.  in  67  pk.  3  qt. 
2          =  No.  pt.  in  1  qt. 

1078  pt.    =  No.  pt.  in  539  qt. 
1  pt. 

Ans.  1079  pt.    =  No.  pt.  in  16  bu.  3  pk.  3  qt.  1  pt. 

2.  Express  (1)   1079  pt.  in  higher  units;  (2)  2  pk.  3  qt.  1  pt. 

in  bushels. 

CONVENIENT  FORM 

(1) 
2)  1079  pt. 

8)  539  qt.  -f-  1  pt-  remaining 
4)  67  pk.  -f  3  qt.  remaining 
16  bu.  -f  3  pk.  remaining 
Ans.  16  bu.  3  pk.  3  qt.  1  pt. 

-    (2) 
8  qt.  1  pt.  =  3.5  qt.  =  -~  pk.  =  .4375  pk. 

2  pk.  3  qt.  1  pt.  =  2.4375  pk.  =  ^      -  bu.  =  .609375  bu.     Ana. 


COMPOUND    DENOMINATE    NUMBERS  231 

P>.  The  three  sides  of  a  triangular  grass  plot  were  :  8  yd.  2  ft. 
10  in.;  12  yd.   1   ft.  0  in.  ;  and  9  yd.   2  ft.   7  in.;  how  far  is  it 

around  the  plot? 

EXPLANATION.  —  First,  adding  the  inches, 
CONVENIENT  FOKM  we  obtain  26  in  ^  2  ft.  2  in.     Write  the  2  in. 

8  yd.  2  ft.  10  in.       in  "inches"  column,  and  add  the  2  ft.  to  the 
12         1         9  numbers  in  the  "feet"  column,  giving  7  ft. 

9  2         7  =  2  yd.  1  ft.     Write  1  ft.  in  "feet"  column 

31  yd.  1ft.    2  in.         ***  nUmberS  ln  "yards 


4.  From  a  vessel  containing  25  gal.  3  qt.  1  pt.  2  gi.  of  oil, 
8  gal.  3  qt.  1  pt.  3  gi.  were  drawn  out;  how  much  oil  remained 
in  the  vessel? 

EXPLANATION.—  3  gi.  can  not  be  taken 
from  2  gi.     But  the  1  pt.  of  the  minuend 

CONVENIENT  FORM  jquals  4  gi.   and  this  added  to  2  gi.  gives 

6  gi.     6  gi.  —  3  gi.  =  3  gi.     1  pt.  from  0  pt. 

25  gal.  2  qt.  1  pt.  2  gi.  can  not  be  taken.     But  1  qt.  is  taken  from 

8  gal.  3  qt.  1  pt.  3  gi.  2qt.,  and  changed   to  pints,  giving  2   pt. 

1fi  0-al   2  nt   1  nt   S  P-i  3    pt.  —  1    pt.  =  1   pt.      1    gal.  =  4    qt.    4 

LO  gal.  2  qt.  1  pt.  6  gi.  ^  +  !  qt.  =  5  qt<      5  qt   _  3  qt   =  2  qt 

Ans.  16  gal.  2  qt.  1  pt.  3  gi.  Finally,  24  gal.  —  8  gal.  =  16  gal.  Prove 
the  work  by  adding  the  remainder  to  the 
subtrahend  and  comparing  the  result  with 
the  minuend. 

5.  A  man  built  an  average  of  45  ft.  8  in.  of  fence  a  day  for 
16  da.  ;  how  much  fence_did  he  build  in  16  days? 

CONVENIENT  FORM 

45  ft.    8  in. 

16  EXPLANATION.  — 

8  in.  X  16  =  128  in.  =  10  ft.  8  in. 
10  ft-    8  in.  45  ft   x  16  =  720  ft 

720 


730  ft.    8  in. 

6.  An  iron  rod,  4  yd.  2  ft.  8  in.  long,  was  cut  into  5  equal 
pieces ;  how  long  was  each  piece? 

EXPLANATIONS  yd.  =12  ft.       12  ft.  +  2  ft. 

5)  4  yd.  2  ft.    8  in.         =  14  f t.     14  ft.  H-  5  =  2,  with  a  remainder  of 
Ans.  2  ft.  11  £ ^  in.       4  ft-     4  ft.  =  48  in.     48  in.  -j-  8  in.  «=  56  in. 

§157.  Exercises  on  Tables. 

1.  Reduce  9  great  gross  to  units. 

2.  How  many  great  gross  are  there  in  79,630  units? 


232  RATIONAL    GRAMMAR    SCHOOL    ARITHMETIC 

3.  A  dealer  bought  3600  lead  pencils  at  $5.00  per  gross.     He 
sold  them  at  25  cents  per  dozen.     What  was  his  profit? 

4.  Bought  2  gr.  foot-rules  at  $.12  a  doz. ;  7  gr.   Spenceriun 
pens  No.  1  at  $.07   a  doz.;  8  gr.  Eagle  pencils  No.  3  at  $.30 
a  dozen.     Find  the  amount  of  my  bill. 

5.  A  stationer   bought  3  gr.  boxes  of  paper,  each  box  con- 
taining 6  reams.     How  many  sheets  did  he  buy? 

6.  A  stationer  sold  3  quires,  20  sheets  of  paper  to  one  man, 
18  quires  to  another,  4  reams,  15  quires  to  another.     How  many 
sheets  did  he  sell  in  all? 

7.  There  are  2   reams,  9   quires   of  paper  in   one  package, 
3  times  as  much  in  a  second  package,  and  7  times  as  much  in 
a  third  package  as  in  the  second  package.      How   much  paper 
is  there  in  the  second  and  third  packages  each? 

8.  In  £28  there  are  how  many  shillings?  how  many  pence? 
how  many  farthings? 

9.  In  18s.  there  are  how  many  pence?    how  many  farthings? 

10.  In  £28  18s.  9d.  3  far.,  there  are  how  many  farthings? 

11.  In  £342  15s.  6d.  there  are  how  many  pence? 

12.  How  many  pounds,  shillings,  and  pence  are  there  in  28,643 
pence? 

13.  How  many  feet  are  there  in  78  yd.?  how  many  inches? 

14.  How  many  inches  are  there  in  78  yd.  2  ft.  6  inches? 

15.  How  many  yards,  feet,  and  inches  are  there  in  2838  inches? 

16.  How  many  ounces  are  there  in  12  cwt.?  in  10  pounds? 

17.  How  many  ounces  are  there  in  12  cwt.  10  Ib.  11  ounces? 

18.  How  many  hundredweight,  pounds,  and  ounces  in  19,360 
ounces? 

There  are  two  classes  of  problems  to  be  solved  in  reducing 
denominate  numbers  to  their  equivalents  in  different  units. 

In  one  class  numbers  expressed  in  larger  units  are  to  be 
expressed  in  smaller  units.  This  is  called  reduction  descending, 
or  reduction  from  higher  to  lower  denominations. 

In  the  other  class  numbers  expressed  in  smaller  units  are  to 
be  expressed  in  larger  units.  This  is  reduction  ascending,  or 
reduction  from  lower  to  higher  denominations. 

Problems  15  and  18  are  examples  of  the  second  class,  and  prob- 
lems 13,  16,  and  17  are  examples  of  the  first  class. 


COMPOUND    DENOMINATE    NUMBERS 

19.  Reduce  160  yd.   1  ft.   9  in.  to  inches. 

20.  Reduce  5781  in.  to  yards,  feet,  and  inches. 

21.  Reduce  8  bu.  2  pk.  5  qt.  to  quarts. 

22.  Reduce  27  T.  18  cwt.  15  Ib.  to  pounds. 

23.  Reduce  23  cu.  yd.  14  cu.  ft.  to  cubic  feet. 

24.  Change  £395  15s.  sterling  to  dollars  and  cents. 

United  States  gold  and  silver  coins  are  .9  pure  gold  or  silver 
and  .1  copper. 

25.  What  weight  of   pure   silver   is   there  in  a  silver  dollar 
weighing  412|  grains? 

26.  The  U.  S.  standard  5  dollar  gold  piece  weighs  129  gr. 
What  is  the  number  of  grains  of  pure  gold  in  the  standard  five 
dollar  gold  piece? 

27.  The    5-cent  piece  weighs  73.16  gr.     .75  of  the  weight  of 
the  coin  is  copper  and   .25  is    nickel.      What  is  the  weight  of 
copper  in  the  5-cent  piece?  of  the  nickel  in  the  5-cent  piece? 

28.  The  eagle  weighs  258  gr.     What  is  the  weight  of  pure 
gold  in  the  eagle? 

29.  How  many  standard  gold  dollars  can  be  coined  from  1  oz. 
of  pure  gold? 

30.  Express  the  following  ratios : 

1  franc   :  $.25;  $.25   :  1  franc; 

1  mark    :  $.25;  $.25  :  1  mark. 

31.  If  .52  oz.  of  gold  is  worth  43s.  4d.,  how  many  ounces  can 
be  bought  for  £35  18s.?      . 

32.  A  boy  laid  by  a  certain  sum  of  money  each  week.     At  the 
end  of  1  yr.  3  mo.  2  wk.  he  had  saved  $88.50.     How  much  did 
he  save  each  week?     (Take  1  mo.  =  4  wk.  2  days.) 

33.  A  man  changed  $350,  half   into  English   and   half   into 
German  money.     How  much  of  each  kind  of  money  did  he  have? 

34.  How  many  feet  are  4  ch.  30  links? 

35.  What  is  the  ratio  of  1760  yd.  to  1  mi.  32  rods? 

36.  The  mast  of  a  ship  was  78  ft.  4  in.  high.     During  a  storm 
.3  of  it  was  broken  off.     How  high  was  the  remaining  piece? 

37.  A  four-sided    field    had  sides  of   the  following  lengths: 
63  ch.  2  rd. ;    49  ch.  14  li. ;    53  ch.  1  rd.  16  li.  and  38  ch.  24  li. 
How  far  is  it  around  the  field? 


234  RATfONAL   GRAMMAR    SCHOOL    ARITHMETIC 

38.  A  man  walked  £  of  the  lengttfof  a  breakwater,  which  was 
1  mi.  243  rd.  5  yd.  long.     How  far  did  he  walk? 

39.  A  knot,    or  geographic  mile,  equals  GOsr,   ft.      What  is 
the  speed  in  common  or  statute  miles  per  hour  of  a  vessel  that 
runs  21  knots  per  hour? 

40.  A  wheel,  12  ft.  in  circumference,  makes  how  many  revolu- 
tions in  1-J-  miles? 

41.  A  telegraph  wire  is  14  mi.  140  rd.  long,  and  is  supported 
by  38(5   poles,  which    are    placed    at    equal    distances    apart,    a 
pole  being  at  each  end  of  the  wire.     How  many  feet  apart  are  the 
poles? 

42.  What  is  the  cost  of  18  A.  120  sq.  rd.  of  land  at  $52  per  acre? 

43.  A  man  owned  14  A.  of  land,  and  sold  1428  sq.  rd.     How 
many  acres  did  he  have  left? 

44.  If  1000  shingles  are  needed  to  cover  100  sq.  ft.  of  roof, 
how  many  shingles  are  required  to  cover  a  roof  40  ft.  long  and 
25  ft.  wide,  at  the  same  rate? 

45.  A  lawn  tennis  court  120  ft.  long  and  85  ft.  wide  is  to  be 
surrounded  by  a  strip  of  sod  15  ft.  wide  at  each  end  and  S  ft. 
wide  at  each  side.     What  will  the  sodding  cost  at  $.35  per  square 
yard? 

46.  Find  the  cost  of  a  half  section  of  land  at  $45  per  acre. 

47.  How  many  square  rods  are  there  in  a  rectangular  field 
24  ch.  45  li.  long  by  16  ch.  34  li.  wide? 

48.  How  many  cubic  inches  are  there  in  a  tank  containing 
120  gal.  ?  how  many  cubic  feet? 

49.  How  many  cubic  feet  are  there  in  a  block  of  stone  5  ft.  x 
4  ft.  x  6  inches? 

50.  How  much  must  I  pay  for  a  board  1C  ft.  long  and  8  in. 
wide,  the  board  being  1  in.  thick  and  lumber  costing  $35  per 
thousand?     (A  board  foot  means  144  sq.  in.  of  surface,  not  over 
1  in.  thick.) 

51.  I  bought  3  boards  12'  long,  6"  wide,  8  boards  16'  long  and 
9"  wide  and  2  boards  10'  long,  12"  wide  and  2"  thick;  what  did 
the  whole  cost  at  $30  per  thousand? 

52.  What  is  the  value  of  a  straight  pile  of  wood  16  ft.  long, 
8  ft.  wide  and  6  ft.  high,  at  $7.50  per  cord? 


COMPOUND    DENOMINATE    NUMBERS  235 

53.  What  is  the  weight  of  a  pile  of  oak  boards  14  ft.  long 
8  ft.  4  in.  wide  and  5  ft.  high,  at  an  average  weight  of  54  Ib.  per 
cubic  foot? 

54.  At  $.75  a  load  of  1  cu.  yd.  what  will  be  the  cost  of  remov- 
ing a  pile  of  earth  60  ft.  long  24  li.  wide  and  1  yd.  high? 

55.  In  one  year  the  quarries  of   Minnesota  yielded  4,000,000 
cu.  ft.  of  sandstone.     What  was  this  worth,  at  $.76  a  perch? 

56.  Find  the  cost  of  building  a  stone  wall  150'  x  10'  x  2-J',  at 
$3.50  a  perch? 

57.  A  bin  12'  x  8'G"  x  5'  is  filled  with  wheat,      What  is  the 
weight  of  the  wheat  at  60  Ib.  per  bushel? 

58.  What  is  the  value  of  a  straight  pile  of  pine  slabs  32'x  7'x  4', 
at  $3.25  per  cord? 

59.  A  grocer  paid  $4.50  for  a  barrel  of  vinegar  (31^-  gal.),  and 
sold  it  at  50  per  quart.     What  was  his  profit? 

60.  A  jug  contained  214  cu.  in.  of  molasses.     How  much  did 
it  lack  of  containing  1£  gallons? 

61.  What  will  1  mi.  of  right  of  way  for  a  railroad  cost  at  $00 
per  acre,  the  width  of  the  right  of  way  being  100  feet? 

62.  What  will  be  the  cost  per  mile  for  railroad  ties  at  $.45 
apiece,  the  ties  being  laid  one  every  2  ft. 

63.  Find  the  cost  of  the  rails  for  1  mi.  of  the  road,  the  rails 
weighing  77  Ib.  per  linear  yard  and  costing  $35  per  long  ton. 

64.  Find  the  cost  of  fencing  1  mi.  on  both  sides  of  the  track, 
placing  posts  costing  $.25  apiece  16  ft.  from  center  to  center,  and 
using  wire  weighing  2160  Ib.  per  mile,  and  30  Ib.  of  staples  @  40. 
The  fence  is  to  be  4  wires  high,  and  labor  costs  210  per  rod  of 
fence. 

65.  Reduce  .865  gal.  to  smaller  units. 

SOLUTION.—  .865  X  4  qt.  =  3.46  qt. ;  .46  X  2  pt.  =  .92  pt. ;   .93  X  4  gi. 
=  3.68  gills.  Ans.  a  qt.  3.68  gills. 

66.  Reduce  .168  gal.  to  smaller  units. 

67.  If  oil  is  $.11  per  gallon,  what  will  be  the  cost  of  tnree  42- 
gallon  barrels  of  kerosene? 

68.  If  a  certain   spring   regularly  yields  25  gal.   daily,  how 
many  barrels  of  the  capacity  of  31i  gal.  would  it  fill  in  20  days? 


69.  A  merchant  bought  16  gal.  3  qt.  of  syrup  at  38$  a  qt.,  and 
sold  27  qt.  for  $12,  18  qt.  for  $9,  and  the  remainder  at  35$  per 
quart.     Did  he  gain  or  lose,  and  how  much? 

70.  Eeduce  2  pk.  7  qt.  1  pt.  to  the  decimal  of  a  bushel. 

SOLUTION.—  1  pt.=  .5  qt.    7.5  qt.=  ~  pk.  =  .9375  pk.     2.9375  pk.  = 
bu.  =  .734375  bushels. 


71.  Reduce  3  pk.  5  qt.  1  pt.  to  bushels. 

72.  How  many  bushels  are  there  in  a  bin  14  ft.  long  7  ft.  6  in. 
wide  and  5  ft.  8  in.  high? 

73.  A  farmer  sold  6  loads  of  corn,  each  load  averaging  36  bu. 
2  pk.  at  35  cents  per  bushel.     What  did  he  receive  for  the  whole? 

74.  A  boy  had  a  bushel  of  hickory  nuts,  and  sold  3  pk.  7  qt. 
1  pt.     What  fraction  of  a  bushel  had  he  left? 

75.  If  9  bu.   of  potatoes  cost  $4.80,  what  is  the  average  cost 
per  peck? 

76.  A  bin  contained  5376.25  cu.  in.  of  rye.     How  much  did  it 
lack  of  containing  3£  bushels? 

77.  How  many  oz.  of  quinine  will  be  required  to  prepare  12 
gross  of  3-grain  capsules? 

78.  A  coal  dealer  buys  two  car-loads  of  coal  each  weighing 
67,200  lb.,  at  $4.50  a  long  ton,  and  sells  it  at  $5.75  a  short  ton. 
What  is  his  gain? 

THE  METRIC  SYSTEM 
§158.  Historical. 

The  metric  system  is  a  decimal  system  of  weights  and  meas- 
ures adopted  by  the  French  Government  soon  after  the  French 
Revolution  of  1789.  The  aim  of  the  system  is  to  base  all  measures 
upon  an  invariable  standard,  and  to  secure  the  simplest  possible 
relations  between  the  different  units  of  the  system. 

The  unit  of  length,  which  is  fundamental  to  the  whole  system, 
is  called  the  meter.  It  was  attempted  to  make  the  meter  1  ten- 
millionth  of  the  length  of  the  part  of  a  meridian  of  the  earth, 
which  reaches  from  the  equator  to  the  pole,  called  a  quadrant  of 
the  earth's  meridian.  The  meridian  of  the  earth  was  measured, 
and  ToWoTnrF  °f  the  quadrant  was  obtained.  A  platinum  bar 
equal  to  this  length  was  very  accurately  cut  and  stored  in  the 
Government  archives  as  the  official  standard  of  reference. 


COMPOUND    DENOMINATE    NUMBERS  237 

Later  measurements  of  the  earth's  meridian  showed  the  former 
length  of  the  meridian  to  be  incorrect.  The  length  of  the  meter 
as  obtained  from  the  erroneous  measures  was,  however,  retained, 
and  the  length  of  this  bar  is  the  standard  meter.  From  it  all  the 
other  units  of  the  system  are  derived. 

The  metric  system  is  used  for  all  purposes  in  France,  and  for 
nearly  all  scientific  purposes  in  Germany,  England,  and  the 
United  States. 

The  length  of  the  meter  is  39.37079  in.,  or  about  1.1  yards. 

§159.  Tables  of  Metric  Measures. 

Fig.  136  shows  a  scale  graduated  along  the  upper  edge  to  16ths 
of  an  inch,  and  along  the  lower  edge  to  lOOOths  of  a  meter,  called 
millimeters. 


I ,,, I .,,.,,  I .,..,.  I ,,..,,  I ,,.  I 

21  31 


.ill.lil.hfl.l.l.l.fl.l.l.l.fl 


FIGURE  136 

Decimal  parts  of  the  standard  units  are  denoted  by  the  Latin 
prefixes,  abbreviated  in  each  case  to  a  single,  small  letter: 

milli,    meaning  1000th  written  m. 
centi,   meaning    100th  written  c. 
deci,     meaning      10th  written  d. 

Multiples   of   the  standard  units  are  denoted  by  the  Greek 
prefixes,  abbreviated  in  each  case  to  a  single,  capital  letter: 

deka,  meaning  10  times  written  D 
hekto,  meaning  100  times  written  H. 
kilo,  meaning  1000  times  written  K. 
myria,  meaning  10000  times  written  M. 

TABLE  OF  MEASURES  OF  LENGTH 

10  millimeters  (mm.)   =  1  centimeter  (cm.)  =  about  .4  in. 

10  centimeters  =  1  decimeter  (dm.)  =  about  4.0  in. 

10  decimeters  =  1  METER  (m.)  =  about  1.1  yd. 

10  meters  =  1  dekameter  (Dm.)  =  about  32.8  ft. 

10  dekarneters  =  1  hektometer  (Hm.)  =  about  328     ft. 

10  hektometers  =  1  kilometer  (Km.)  =  about  .62  roi 

10  kilometers  =  1  myriameter  (Mm.)  =  about  6.21  mi 


238 


RATIONAL    GRAMMAR    SCHOOL    ARITHMETIC 


TABLE  OF  SURFACE  MEASURE 


100  sq.  millimeters  (mm2.) 
100  sq.  centimeters 
100  sq.  decimeters 
100  sq.  meters 
100  sq.  dekameters 
100  sq.  hektometers 


=  1  sq.  centimeter  (cm2.) 
=  1  sq.  decimeter  (dm2.) 
=  1  sg.  MKTKR  (in-.) 
=  1  sq.  dekameter  (Dm-'.) 
=  1  sq.  hektometer  (Hm2.) 
=  1  sq.  kilometer  (Km'2. ) 


SQ.INCH 

SQ. 
CENTI- 
METER 

FlGUKE  137 


The  cm2.    =        .155sq.  in. 
Them2.      =    10. 764  sq.ft. 
The  Km2.  ==  247.114  A. 
1  m2  =  1  cen tare  (ca.) 

TABLE  OF  LAND  MEASURE 

100  centares  =  1  are  (pronounced  air)  (a.) 
100  ares  =  1  hektare  (Ha.) 


TABLE  OF  MEASURES  OF  VOLUME 

The  standard  unit  of  volume  is  the  cubic  meter  =  35.314  cu. 
ft.  =  about  1.2  cubic  yards. 

1000  cubic  millimeters  (mm3.)  =  1  cu.  centimeter  (cm3.) 
1000  cubic  centimeters  =  1  cu.  decimeter  (dm3.) 

1000  cubic  decimeters  =  1  cu.  meter  (m3. ) 

etc.,  etc. 


TABLE  OF  MEASURES  OF  CAPACITY 


The  standard  unit  of  capacity  is  the  liter  (leeter).     It  is  equal 
to  1  cu.  decimeter,  and  equivalent  to  .908  dry  quarts. 

10  milliliters  (ml.)  =  1  centiliter  (cl.) 


10  centiliters 
10  deciliters 
10  liters 
10  dekaliters 
10  hektoliters 


=  1  deciliter  (dl. ) 
=  1  liter  (1.) 
=  1  dekaliter  (Dl.) 
=  1  hektoliter  (HI) 
=  1  kiloliter  (Kl.) 


TABLE  OF  MEASURES  OF  WEIGHT 


The  standard  unit  of  weight  is  the  gram,  which  is  the  weight 
of  1  cu.  centimeter  of  distilled  water  at  its  temperature  of  greatest 
density  (39.1°  F.). 


COMPOUND    DENOMINATE    NUMBERS 


231) 


10  milligrams  (mg. ) 
10  centigrams 
10  decigrams 
10  grams 
10  dekagrams 
10  hektograms 
10  kilograms 
10  myriagrams 
10  quintals 


1  centigram  (eg.) 
1  decigram  (dg.) 
1  gram  (g.) 
1  dekagram  (Dg.) 
1  hektogram  (Hg.) 
1  kilogram  (Kg.) 
1  myriagram  (Mg). 
1  quintal  (Q.) 
1  metric  ton  (T.) 


1  centigram  =         .15432  grain 
1  gram  =      15.432  grains 

1  kilogram     =       2.20462  Ib. 
1  metric  ton  =  2204.621  Ib. 


§160.  Metric  and  II.  S.  Equivalents. 

The  equivalents  will  be  of  assistance  in  changing  the  metric 
to  the  common  system  of  measures. 


1  m.     =  39.37  in. 
lKm.=      .6214  mi. 

1m2.  =  1.196  sq.  yd. 
1  Km2.  =  .3861sq.ini. 
1  are  =  .0247  A. 

1  ra3.  =  1.308  cu^  yd. 
1  stere  =  .2759  cord 


1  liter 
1H1. 


=  j    1.0567  liquid  qt. 
"  (      .9081  dry  qt. 
__  j    2.8376  bu. 
(  26.417  liq.  gal. 


1  g.  =  15.432  grains 

IKg.  =    2.2046  Ib.  avoir. 

1  metric  ton  =    1.1023  T. 


1  mile  =  1.6093  Km. 
1  yard  =    .9144  m. 

1  square  yard  =  .8361  in2. 
1  square  mile  =  2.59  Km2. 
1  acre  =  40. 47  a. 


1  cubic  yard 
1  cord 


=    .7645ms. 
=  3.624  st. 


61.022cu.  in.  =  11. 

1  liquid  qt.  =    .9436  1. 

1  dry  qt.  =  1.101  1. 

1  bushel  =    .3524  HI. 

1  grain  =  .0648  g. 

1  Ib.  avoir.  =  .4536  Kg. 

1  T.  =  .9072  T. 


PROBLEMS   WITH    THE    METRIC    UNITS 

1.  Take  a  smooth  lath,  or  plane  one,  lay  a  straight  strip  of 
paper  beside  the  lower  edge  of  the  metric  rule,  Fig.  130,  page  237, 
and  with  a  sharp  pencil  transfer  the  graduations  from  the  rule  to 
the  strip  of  paper,  and  then  from  the  strip  of  paper  to  the  lath. 


240  RATIONAL   GRAMMAR   SCHOOL   ARITHMETIC 

Continue  the  centimeter  graduation  marks  along  your  lath  until 
you  get  a  meter  stick  (=  100  centimeters).  How  does  this  meter 
stick  compare  in  length  with  a  yard  stick? 

2.  Explain  which  digit  in  6.8752  m.  stands  for  m. ;  which  for 
dm. ;  which  for  cm.,  and  which  for  millimeters. 

3.  Find  by  measurement  the  length  and  the  width  of  your 
schoolroom  in  meters,  and  write  the  result  of  your  measurements 
in  meters  and  decimals  of  a  meter. 

4.  How  many  centimeters  long,  wide,  and  thick  is  your  arith- 
metic? 

5.  Express  the  length  of  your  pencil  in  centimeters. 

6.  A  sidewalk  is  2112  m.  long,  how  many  kilometers  long  is  it? 

7.  What  is  the  cost  of  158  cm.  of  ribbon  at  $.45  a  meter? 

8.  How  many  steel  rods,  each  16  cm.  long,  can  be  cut  from 
a  rod  7.68  m.  long? 

9.  From  a  piece  of  wire,  98.52  m.  long,  a  piece  .047  Km.  was 
cut.     How  long  was  the  remainder? 

10.  A  bicycle  track  is   .807  Km.  long.      How  many  meters 
would  one  go  in  riding  around  the  track  5  times? 

11.  Express  1   Km2,  in  square  dekameters;  in  square  meters. 

12.  Find  the  number  of  square  meters  in  the  surface  of  your 
schoolroom.     Express  the  same  in  square  dekameters. 

13.  Express  the  area  of  your  desk  in  square  decimeters;  in 
square  inches.     How  do  the  two  compare? 

14.  Using  the  decimeter  as  a  measure,  find  the  number  of 
square  centimeters  in  the  surface  of  your  geography. 

15.  Express  the  same  in  square  decimeters;  in  square  inches. 

16.  Find  the  area  of  the  door  in  square  meters.    What  part  of 
an  are  is  this  area? 

17.  A  certain  plot  of  ground  contains  20  m2.   How  many  square 
feet  in  the  plot? 

18.  With  the  aid  of  your  meter  stick,  lay  off  a  square  meter 
upon  the  floor.     Within  this  area,  lay  off  a  square  yard.     Note 
results. 

19.  With  the  aid  of  your  decimeter  measure,  draw  a  square 
decimeter.     In  the  corner,  draw  a  square  centimeter.     How  many 
square   centimeters  could  be  drawn  within  the  square  decimeter? 


COMPOUND    DENOMINATE    NUMBERS  241 

20.  What  is  the  area  of  a  floor  8.4  m.  long  and  5  m.  wide? 

21.  A  lot  is  7.64  m.  wide  and  .033  Km.  deep.     What  is  the 
area? 

22.  What  is  the  value  of  a  piece  of  land  8  Dm.  long  and  6. 5  Dm. 
wide,  at  $32  an  are? 

23.  How  many  ares  in  a  street  1.5  Dm.  wide  and  2.48  Km. 
long? 

24.  How   many  square  meters   in  a  floor  700  cm.  long  and 
500  cm.  wide? 

25.  A  man  had  3  pieces  of  land  as  follows:  8.4  Ha.;  3846 
m2.,    and    2.5    Km2.     How  many  hektares   of   land   are    there 
in  all? 

26.  20  liters  equal  how  many  centiliters?  what  part  of  a  Dl.  ? 
of  a  KL?     Find  the  difference  between  12  1.  and  12  quarts. 

27.  Using  your  decimeter  measure  as  a  basis,  model  a  cubic 
decimeter  or  liter. 

28.  Using  your  square  centimeter  as  a  basis,  model  a  cubic 
centimeter.     What  is  the  relation  of    a    cubic  centimeter  to  a 
gram?     How  many  cubic  centimeters  in  a  liter? 

29.  How  many  cubic  meters  are  there  in  the  volume  of  your 
schoolroom? 

30.  What  is  the  difference  between  an   ordinary  ton  and  a 
metric  ton  (tonneau)? 

31.  A  gram  is  the  weight  of  1  cm3,  of  distilled  water.     What 
is  the  weight  in  grams  of  1  1.  of  distilled  water? 

32.  A  liter  of  water  weighs  nearly  2|  Ib.     What  is  the  weight 
of  6  1.  of  water  in  pounds?  in  grams? 

.33.  A  package  of  silver  weighs  2.58    grams.          What  is  its 
weight  in  grains? 

34.  How  many  pounds  in  1  Q.?  in  5  quintals? 

35.  How  many  2  gr.  capsules  will  5  g.  of  quinine  fill? 

36.  What  would  be  the  cost  of  the  quinine  at  10^  per  dozen 
capsules? 

37.  What  is  the  cost  of  5500  Kg.  of  coal  at  $6.50  per  ton? 

38.  What  is  the  cost  of  2  Q.  of  sugar  at  $.08£  a  kilogram? 

39.  If  a  book  weighs  3.2  Dg. ,  how  many  such  books  will  weigh 
1.792  kilograms? 


JtVAAlUiX  ALi     \xtlAaLJU.  All, 


40.  A  box  contains  2500  packages  of  quinine,  each  package 
weighing  1.25  g.     How  many  hektograms  does  the  contents  of 
the  box  weigh? 

41.  A  vessel,  18  cm.  long,  14  cm.  wide,  and  22  cm.  high,  is 
filled  with  lead,  which  weighs   11.35  times  as   much  as   water. 
What  is  the  weight  of  the  lead?  (1  cu.  ft.  of  water  weighs  62.5  Ib.) 


PEECENTAGE  AND  INTEREST 
§161.  Percentage. 

ORAL  WOKK 

1.  $2   equals  what  part  of  $8?     9  bu.  equals  what  part  of 
12  bu.?     8  hr.  equals  what  part  of  24  hr.?     $6  equals  what  part 
of  $9? 

2.  To  how  many  hundredths  of  a  number  are  the  following 
fractional  parts  of  that  number  equal  : 

19    1  9    3  9     1  9    2  9    3  9     1  ?    2  9    3  ?      1V      39      79      19      39      49 
%•    ?•    -4-     ,T-     3-     3-    ¥*    -§'    f*    TIT-    TO-    TO-     -0V-     W    h' 

3.  Express  the  following  as  hundredths: 

A;  A;  iV»  -fut  i;  !;  iV>  i%>  4;  1;  I;  IT- 

DEFINITIONS.  —  The  words  per  ce?i£  mean  hundredth,  or  hundredths. 
The  sign,  "  $,"  is  a  short  way  of  writing  jper  cent,  or  hundredths; 
thus,  2$,   8$,  38i%,  mean  -,§3,   jfo,  |^|,  and  are  read,  "2  per  cent," 


8  percent,"  "33£  percent." 

The  number  (as,  2,  8,  33£),  written  before  the  sign,  "%,"  is  called 
the  rate  per  cent. 

It  is  well  to  recall  that  we  have  the  following  ways  of  writing  such 
numbers  as  6  per  cent:  (1)  6  per  cent;  (2)  6^  ;  (3)  6  hundredths;  (4)  r$ff, 
and  (5)  .06.  The  sign,  "  %,"  has  the  numerical  value  of  ifa,  or  .01. 

4.  Eeferring  to  the  table  (§7,  p.  15),  find  what  per  cent  of  the 
total  number  of  animals  store  food  for  winter.  Answer  other 
similar  questions  on  the  table. 

WRITTEN    WORK 

Review  §85,  pp.  129-131. 

NOTE.  —  Whenever  you  can  answer  a  problem  orally,  do  so.  Form  the 
habit  of  using  your  pencil  only  when  it  is  necessary. 

1.  In  a  schoolroom  containing  40  pupils,  f  of  the  pupils  were 
girls.  How  many  girls  were  there? 


PERCENTAGE    AND    INTEREST  243 

2.  In   the  last  problem,  'what   per  cent  of  the  pupils  were 
girls? 

3.  In  the  month  of  September,  12  da.  were  cloudy.     What  per 
cent  of  the  clays  were  cloudy? 

4.  Express  the  equivalents  of  the  following  fractions  as  per 
cents,  or  hundredths  : 

I;  i;  tt;*;  4;  *;  A;  T"*;  A;  H- 

5.  In  a  sample  of  soil  weighing  37  oz.,  18  oz.  were  sand.    What 
per  cent  of  the  soil  was  sand? 

0.  A  sample  stick  of  timber,  weighing  19  lb.,  contained  3  Ib. 
of  water.  What  per  cent  of  the  weight  of  the  timber  was  due  to 
the  water  it  contained? 

SOLUTION.  —  (1)  Annex  zeros  after  the  numerator  and  divide  thus: 
19)2.00        But  .11^  =  11  iV#-     Why? 


7.  What  percent  of  a  yard  is  1  ft.?    9  in.?    15  in.?   28  in.? 
35  in.?    1  inch? 

8.  The  first  number  of  these  pairs  is  what  per  cent  of  the 
second  : 

(1)  8  of  150?       (4)       4|    of    GO?       (7)  32.91  of  263.28? 

(2)  13  of  700?       (5)    27J    of  600?       (8)  J         of  J? 
(3)7Jof    12?       (6)  38.45  of  769?       (9)  |         of  12J? 

9.  Change  the  following  per  cents  to  their  fractional  equiva- 
lents (fractions  in  their  lowest  terms)  : 

5%;  8%;  8J%;  12J%  ;  12%;  16%;  16|;  22J%  ; 


10.   Change  to  their  decimal  equivalents  the  following: 

7%  ;  6i%  ;  83i%  ;  87^%  ;  \\%  ;  2|%  ;  |%  ;  TV  of  1%  ;  T^  of  1%. 


11.  At  the  beginning  of  a  school  year,  the  lung  capacity  of  a 
boy  was  161  cu.  in.     By  the  middle  of  the  year  it  was  7%  larger. 
How  many  cubic  inches  had  his  lung  capacity  increased? 

12.  The  lung  capacity  of  a  boy  at  the  beginning  of  the  year 
was  160  cu.  in.,  and  at  the  middle  of  the  year  it  was  166.4  cu.  in. 
What  was  the  per  cent  of  increase? 


244  RATIONAL    GRAMMAR    SCHOOL    ARITHMETIC 

13.  The  standing  of  the  several  clubs  in  the  National  League 
during  one  season  was  determined  from  this  table : 


CLUB 
Pittsburg  

GAMES  WON 
90 

GAMES  LOST 
49 

PER  CENT  WON 
64.7$ 

Philadelphia  .... 
Brooklyn  

83 
79 

57 
57 

St.  Louis  

76 

64 

69 

69 

Chicago  

53 

86 

New  York  
Cincinnati  

52 

52 

85 

87 

Find  for  each  club  what  per  cent  of  the  total  number  of  games 
played  were  won.  Carry  computations  to  the  first  decimal  place, 
as  indicated  for  Pittsburg. 

14.  What  per  cent  of  the  farm,  Fig.  7,  p.  11,  is  the  cornfield? 
the  wheatfield?   the  meadow?    the  south  oat  field?    the  pasture? 
the  lot  occupied  by  the  house  and  grounds? 

15.  60%  of  the  value  of  a  mill  is  $5400;  what  is  the  value  of 
the  mill? 

16.  3%   of  a  school  of  1200  children  were  absent;  how  many 
children  were  absent? 

17.  I  pay  12%  of  the  value  of  the  property  I  occupy  as  rent 
every  year.     My  rent  is  $240  a  year;  what  is  the  value  of  the 
house? 

18.  A  house  was  damaged  by  fire,  and  an  insurance  company 
paid  the  owner  $840  damages,  which  was  40%  of  the  original  cost 
of  the  house ;  what  was  the  original  cost? 

§162.  Algebra. 

ORAL   WORK 

1.  50  equals  what  per  cent  of  100?  of  200?  of  150?  of  500? 

2.  2  equals  what  per  cent  of  4?  of  8?  of  6?  of  20? 

3.  Any  number  equals  what  per  cent  of  a  number  twice  as 
large?  4  times  as  large?  3  times  as  large?  10  times  as  large? 

4.  x  equals  what  per  cent  of  2x?  of  4#?  of  3#?  of  10#? 

5.  70  equals  what  per  cent  of  14«?  of  28a?  of  21a?  of  70a? 

6.  13  equals  what  per  cent  of  17?  of  28?  of  45?  of  55? 


PERCENTAGE    AND    INTEREST  245 

7.  13#  equals  what  per  cent  of  17#?  of  28z?  of  45:r?  of  55x? 

8.  a  equals  what  per  cent  of  ?/?  of  z?  of  w?  of  p? 


100- 


WRITTEN    WORK 

1.  What  is  2%  of  $200?  of  $375?     How  is  2%  of  any  number 
of  dollars  found? 

2.  How  is  2%  of  $a  found?    What  is  2%  of  $a?  of  $£?  of  $z? 
of  $Gz? 

3.  How  is  5%   of  any  number  found?     What  is   5%  of  the 
number  a?  of  £?  of  a?  of  s?  of  82? 

4.  How  is  12^%  of  any  number  found?     What  is  12£%   of 
a?  of  x?  of  lOaj? 

5.  II  ow  is  any  per  cent  of  a  number  found? 

G.  How  can  you  express  a  as  hundredths?  x,  as  hundredths? 
9z?  12?/?  452? 

7.  Express   these  numbers  as  hundredths:  16;  20;  a;  x\  m\ 
IQx. 

8.  What  is  r%  of  160?  of  350?  of  «?  of  W  of  6'?  of  2wi? 

DEFINITIONS.  —  The  result  of  finding  a  given  per  cent  of  any  amount, 
or  number,  is  called  the  percentage.  The  amount,  or  number,  on  which 
the  percentage  is  computed  is  often  called  the  base. 

9.  Calling  the  percentage,  p,  the  rate  per  cent,  r,  and  the 
base,  #,  show  by  an  equation  how  to  find  p,  from  #  and  r. 

10.  Eeplacing  the  symbols  in  your  equation  by  the  words  for 
which  they  stand,  translate  the  equation  into  the  common  lan- 
guage of   percentage.     This   translation,   properly  made,  is  the 
fundamental  principle  of  percentage. 

PRINCIPLE.  —  The  percentage  equals  the  product  of  the  base  and 
rate  divided  ly  100,  or  more  briefly, 

(i)     P  -  &• 

All  of  percentage  is  contained  in  this  equation,  called  a 
formula,  because  it  formulates  a  law. 

Multiplying  both  sides  of  the  formula  by  100,  we  have 
(A) 


246  RATIONAL    GRAMMAR    SCHOOL    ARITHMETIC 

Now  divide  both  sides  of  this  equation  by  r,  and  write  the 
second  number  (see  §74,  p.  103)  first,  and  obtain  (note  that 

br         7  \ 
-,T  =   #), 

(II)        I  =  UL2£!. 

11.  Translate  (II)  into  words.     How  would  yon  find  the  base 
(b)  if  the  percentage  (p)  and  rate  (r)  were  given? 

12.  Divide  both  sides  of  (A)  by  Z>,  write  the  second  number 
first,  and  make  a  rule  for  finding  r  when  p  and  b  are  given. 

§163.  Gain  and  Loss. 

1.  A  merchant  paid  $10  for  a  suit  of  clothes.     At  what  price 
must  he  sell  the  suit  to  gain  25  per  cent? 

2.  A  stationer  pays  $1.80  a  dozen  for  blank  books,  and  retails 
them  at  a  profit  of  66f  %.     At  what  price  per  book  does  he  sell 
them? 

3.  A  grocer  buys  eggs  at  wholesale  at  100  a  doz.,  and  retails 
them  at  a  profit  of  30%.     Supposing  there  is  no  loss  due  to  break- 
age or  spoiling,  at  what  price  per  dozen  does  he  sell  them? 

4.  Allen  paid  $1  for  a  sled,  and  sold  it  at  a  loss  of  20%.    How 
much  did  he  receive  for  the  sled? 

5.  From  a  box  containing  200  oranges,  80%  were  sold  in  one 
day.     How  many  oranges  were  sold? 

6.  What  is  the  percentage  on  $640  at  20%?  at  28%?  at  35%? 
at  6i%?  at  124%?  afc  87|%? 

7.  A  farm,  costing  $4400,  was  sold  at  a  gain  of  18%  ;  for  how 
much  did  the  farm  sell? 

8.  A  farm  sold  for  $5192,  which  was  18%  more  than  was  paid 
for  it.     How  much  was  paid  for  it? 

SOLUTION.— The  18%  here  means  18%  of  what  was  paid  for  the  farm 
(the  cost  price).  $5192  is  then  equal  to  what  per  cent  of  the  cost?  The 
statement  may  be  written  thus : 

1.18  times  the  cost  price  =  $5192, 

or,  more  briefly,  thus:    l.lSa;  =  $5192.     Hence,  x  =     .      .     Divide  and 

l.lo 
find  the  value  of  x. 

9.  A  center  fielder  threw  a  ball  90  ft.,  which  was  12%  farther 
than  the  third  baseman  threw  it.    How  far  did  the  third  baseman 
throw  it? 

SUGGESTION.— First  answer  the  question,  12%  of  what? 


PERCENTAGE    AND    INTEREST  247 

10.  A  boy  bought  oranges  at  the  rate  of  5  for  3^,  and  sold 
them  at  the  rate  of  3  for  5<p ;  what  was  his  rate  per  cent  of  gain 
or  loss? 

11.  A  merchant  in  shipping  150  crates  of  eggs  lost  25  crates 
by  freezing.     What  per  cent  did  he  lose? 

12.  Hats,  costing  $2.75,  sold  for  $3.50.     What  was  the  per 
cent  of  profit? 

13.  An  automobile,  costing  $750,  sold  for  $1250.     What  was 
the  per  cent  of  profit? 

14.  In  a  certain  battle,  32,000  men  were  engaged,  and  35%  of 
all  engaged  lost  their  lives.     How  many  lost  their  lives? 

15.  A  horse,  costing  $88,  sold  at  a  loss  of  40%.     For  how 
much  did  the  horse  sell?     (40%  of  what?) 

10.  I  paid  $35  for  a  bicycle,  and  sold  it  the  next  season  for 
60%  less  than  I  paid  for  it.     What  was  the  selling  price? 

17.  A  cow  gave  13  Ib.  of  milk  Nov.  1,  and  15.  G  Ib.  on  Nov.  15. 
What  was  the  rate  per  cent  of  gain  in  her  daily  yield  of  milk? 

18.  Make  similar  problems  on  the  table,  page  21. 

19.  The  increase  in  population  in  one  .year  in  a  certain  town 
was  2000,  which  was  G£%  of  the  population  at  the  end  of  the 
year?     What  was  this  population? 

20.  The  death  rate  the  same  year  was  1.8%  of  this  popu- 
lation.    How  many  deaths  occurred? 

21.  A  man  lost  27%  of  the  books  of  his  library  by  fire.     He 
lost  540  books.     How  many  books  did  his  library  contain  before 
the  fire? 

22.  Anthracite  coal  cost  $8.50  a  ton  in  Nov.,  and  rose  15%  in 
1  mo.     What  was  the  price  of  anthracite  coal  after  the  rise  in 
price? 

23.  I  sold  a  lot  for  $600,  at  a  loss  of  20%.     What  did  the 
lot  cost? 

24.  Find  the  cost  of  a  piano  which  sold  for  $187.50  at  a  loss 
of  25%. 

25.  I  bought  a  tennis  outfit  for  $35,  and  sold  it  at  a  loss  of 
15%.     Find  the  selling  price. 

26.  Carpet,  marked  at  $1.25,  sold  at  a  reduction  (discount) 
of  6f  %.     Find  the  selling  price. 


248  RATIONAL   GRAMMAR   SCHOOL   ARITHMETIC 

27.  A  man  lost  20%  of  his  money  and  after  losing  10%  of  the 
remainder  had  $3,600  left.     How  much  had  he  at  first? 

28.  A  dealer  sells  a  cask  of  30  gal.  of  oil,  and  receives  830. 
In  delivering  it  4.5  gal.  were  spilled.     What  per  cent  was  spilled, 
and  how  much  should  the  dealer  refund? 

29.  Goods  damaged  by  fire  were  sold  at  a  loss  of  40%.     The 
amount  received  was  $555.     What  was  the  original  cost? 

30.  A  suit  of  clothes  was  marked  at  $45,  which  was  50%  more 
than  the  cost.     It  sold  at  a  reduction  of  20%  from  the  marked 
price.     What  was  the  per  cent  of  profit  on  the  original  cost  of  the 
suit? 

§164.  Meteorology. 

The  number  of  clear,  cloudy,  partly  cloudy,  and  rainy  or  snowy 
days  for  the  first  4  mo.  of  1903,  at  Chicago,  are  here  tabulated: 

1903  PrFAR  PLOTTDY  PARTLY        RAIN  OB 

CLEAB  CLOUDY          SNOW 

Jan 9  15  7  9 

Feb 10  11  7  9 

Mar 1£  14  5  11 

Apr 10  13  7  13 

1.  What  per  cent  of  the  number  of  days  of  Jan.  were  clear? 
cloudy?  partly  cloudy?  rainy  or  snowy? 

2.  Answer  similar  questions  for  Feb. ;  for  March;  for  April. 

3.  What  per  cent  of  the  total  number  of  clear  days  of  1903  to 
May  1st  fell  in  Jan.?  in  Feb.?  in  March?  in  April? 

4.  Answer  similar   questions   for   cloudy,  partly  cloudy   and 
rainy,  or  snowy  days. 

5.  The  highest  velocities  in  miles  per  hour  of  the  wind  in 
Chicago  for  each  of  the  12  mo.  (beginning  with  Jan.)  of  a  certain 
year  were:    50;    48;    57;    70;    54;    45;    56;  47;  52;  58;  52;  58. 
What  per  cent  of  the  highest  velocity  for  April  was  the  highest 
velocity  for  Jan.?  for  Feb.?  for  each  of  the  remaining  months? 

6.  The  total  movement,  in  miles,  of  the  wind  in  Chicago  for 
each  of  the  12  mo.  of  a  certain  year  was:  in  Jan.  12,736;  10,279; 
13,999;  12,820;  12,356;  10,900;  11,231 ;  9,839;  11,834;  13,148; 
13,583;  15,476.     What  per  cent  of  the  total  wind  movement  in 
Dec.  was  the  total  movement  for  each  of  the  other  months? 


PERCENTAGE    AND    INTEREST  249 

The  table  below  gives  the  height  in  feet  (elevation),  above 
sea  level  and  the  total  wind  movement  in  miles  for  an  entire  year, 
of  a  number  of  important  weather  signal  stations.  Block  Island 
is  on  the  Rhode  Island  coast,  and  Mount  Tamalpais  is  a  mountain 
station  near  San  Francisco.  Roseburg  (Oregon)  is  the  quietest 
place,  as  to  wind  movement,  reported  in  the  United  States: 

STATION  ELEVATION      WIND  MOVEMENT 

Mount  Tamalpais 2,375  163,203 

Block  Island 26  153,838 

Chicago 823  145,193 

Cleveland 762  128,566 

New  York 314  127,267 

Buffalo 767  125,042 

Boston 125  93,755 

Philadelphia 117  95,319 

St.  Louis  567  84,482 

New  Orleans 51  74,299 

Louisville 525  70,396 

Washington 112  63,629 

Roseburg  (Oregon) 518  30,471 


7.  What  per  cent  is  the  wind  movement  of  Chicago  of  that  of 
each  of  the  other  places  in  the  list? 

8.  What  per  cent  is  the  elevation  of  Chicago  of  that  of  each 
of  the  other  places? 


9.  100  green  oak  leaves  weighed  100    grams.      When   thor- 
oughly dried  they  weighed  39.5  grams.     The  dried  leaves  were 
then  burned,  and  the  ash  weighed  2.4  grams.     What  per  cent  o'f 
the  leaves  was  dry  solid?  mineral  matter  (ash)? 

10.  100  green  elm  leaves  weighed  60  grams.    38£%,  by  weight, 
was  dry  solid,  and  2f  %  was   mineral  matter.      What  was  the 
weight  of  the  dry  solid?  of  the  mineral  matter? 

11.  A  school  garden,  15'x40',  was  seeded  as  follows:  25%, 
potatoes;  16f%  of  the  remainder,  peas;  20%  of  the  remainder, 
beans;   25%  of  the  remainder,  lettuce;    33£%  of  the  remainder, 
radishes;  20%  of  the  remainder,  parsley,  and  the  rest  in  beets. 
How  many  square  feet  were  seeded  to  beets? 


250 


RATIONAL   GRAMMAR    SCHOOL   ARITHMETIC 


12.  The   precipitation    (rainfall)    at    Chicago,    in   inches,   for 
Jan.,  Feb.,  Mar.,  and  Apr.  for  1903  was  1.09,  3.03,  1.67,  and 
3.77,  and  the  averages  for  these  same  months  for  33  yr.  are  2.05, 
2.11,  2.52,  and  2.73.     Find  the  rate  per  cent  of  excess,  or  defi- 
ciency, for  each  of  the  four  months  of  1903. 

13.  Problems  may  be  made  on   the  following  table   of    data 
relating  to  Chicago  weather: 


TEMPERATURE 

PRECIPITATION 

WEATHER 

1902 

Monthly 
Mean 

Mean  for 
31  Years 

In  Inches 

Average 
for  32  Yr. 

Clear 
Days 

Fair 
Days 

Cloudy 
Days 

Jan.     . 

25.2 

23.8 

.66 

2.08 

10 

14 

7 

Feb.     . 

20.8 

25.9 

1.53 

2.30 

12 

9 

7 

Mar.     . 

38.6 

34.3 

4.16 

2.56 

6 

13 

12 

Apr.     . 

46.4 

46.4 

2.26 

2.70 

10 

12 

8 

May     . 

59.0 

56.5 

5.08 

3.59 

10 

19 

2 

June    . 

64.2 

66.6 

6.45 

3.79 

7 

16 

7 

July    . 

72.4 

72.3 

5.78 

3.61 

12 

13 

6 

Aug.    . 

68.4 

71.1   • 

1.44 

2.83 

15 

10 

6 

Sept.    . 

60.8 

64.4 

4.83 

2.91 

9 

8 

13 

Oct.  .    . 

55.2 

53.1 

1.45 

2.63 

14 

10 

7 

Nov.    . 

47.0 

38.5 

2.03 

2.66 

8 

13 

9 

Dec.     . 

26.5 

29.0 

1.90 

2.71 

7 

8 

16 

§165.  The  Almanac. 

NOTE. — In  this  section  "length  of  the  day"  means  the  duration  of  day- 
light, or  the  time  interval  from  sunrise  to  sunset. 

1.  On  Jan.  1,  at  Chicago,  the  sun  rose  at  7  hr.  29  min.,  and  set 
at  4  hr.  38  min.,  and  on  July  1,  it  rose  at  4  hr.  28  inin.,  and  set 
at  7  hr.  39  min.     The  length  of  the  day  on  Jan.  1  equals  what 
per  cent  of  the  length  on  July  1? 

2.  On  Jan.  1,  the  length  of  the  day  in  St.  Louis  was  103.8%, 
and  in  St.  Paul  96.4%,  of  the  length  of  the  same  day  in  Chicago. 
Find  the  day's  length  in  each  of  these  cities. 

3.  On  Julyl,  in  the  same  places,  the  day's  lengths  were,  respect- 
ively, 97.9%  and  102.2%  of  its  length  in  Chicago.    Find  the  day's 
length  in  each  of  these  cities  on  this  date. 

4.  Make  and  solve  similar  problems  for  the  calendar  pages  of 
a  common  almanac. 


PERCENTAGE    AND    INTEREST  251 

5.  On  the  first  of  Feb.,  of  March,  of  April,  of  May,  of  June, 
of  July,   of  Aug.,  of  Sept.,  of  Oct.,  of  Nov.,  and  of  Dec.,  the 
day's  lengths  in  Chicago  were  108%,  122.3%,  137.8%,  152.6%, 
163.2%,  163.5%,  157.9%,  144.6%,  129.2%,  132.5%,  and  102.6% 
of  the  day's  length  on  Jan.  1.     Find  the  day's  length  on   the 
dates  named. 

6.  On  vertical  lines,  one  for  each  month,  plot  these  rates  to 
scale  and  draw  through  the  points  a  smooth  freehand  curve. 

7.  Similar  problems  may  be  obtained  from  the  calendar  pages 
of  a  common  almanac. 

8.  From  the  almanac  find  what  per  cent   the   period  from 
"moon  rises"  to  ''moon  sets"  on  the  first  of  some  month  is  of  the 
same  period  on  the  15th  or  20th  of  the  same  month. 

9.  Find  what  per  cent  the  time  from  full  moon  to  new  moon 
is  of  the  time  from  new  moon  to  first  quarter;  from  new  moon  to 
second  quarter,  or  full  moon. 

§166.  Geography. 

Compute  these  per  cents  to  one  place  of  decimals. 

1.  Refer  to  the  table  of  p.  44,   and  find  what  per  cent  of  the 
population  (1900)  of  your  state  was  in  elementary  and  secondary 
(high)  schools. 

2.  Compare  the  result  of  problem  1   with  the  corresponding 
results  for  any  other  states. 

Data  for  the  following  problems  will  be  found  on  pp.  44  and  53. 

3.  What  per  cent  of  the  area  of  the  United  States  (without 
outlying  territory)  is  the  area  of  the  North  Atlantic  division?   of 
the  Western  division? 

4.  What  per  cent  of  the  total  population  of  the  United  States 
(continental)  is  the  population  of  the  North  Atlantic  division? 
of  the  Western  division? 

5.  Find  what  per  cent  the  area  and  population  of  your  state  is 
of  the  area  and  population  of  the  division  to  which  your  state 
belongs. 

6.  Find  the  per  cent  of  increase  of  population  of  your  state 
from  1890  to  1900. 


252  RATIONAL    GRAMMAR   SCHOOL   ARITHMETIC 

7.  Find  the  per  cent  of  increase  of  population  of  the  (conti- 
nental) United  States  during  the  same  time. 

8.  Find  the  per  cent  of  increase  in  population  from  1890  to 
1900  of  the  division  to  which  your  state  belongs. 

9.  The  following  table  shows  the  territorial  growth  of   the 
United  States  and  the  dates  of  acquisition.     Find  the  per  cent 
of  increase  of  each  addition  on  the  date  of   acquiring  to   1899 
inclusive  : 


ACQUISITION  j  DATE 

Original  territory  ................     1783  827,844 

Louisiana  purchase  ...............     1803  1,  182,  752 

Florida  ..........................     1819  59,268 

Texas  ...........................     1845  371,063 

Mexican  purchase  ................     1848  522,568 

Texas  purchase  ..................     1850  96,707 

Gadsden  purchase  ...............     1853  45,535 

Alaska  ..........................     1867  590,884 

Hawaii  ..........................     1898  6,449 

Porto  Rico  ......................  ^1  3,600 

Philippine  Islands  ................  j>  1899  114,000 

Guam  ............................  J  200 

Isle  of  Pines  .....................     1899  882 

10.  The  area  in  millions  of  square  miles  of  the  surface  of  the 
earth  is  148.18;  of  this  surface  108.77  is  water,  and  39.41  land. 
Find  what  per  cent  of  the  surface  of  the  earth  is  water;  land. 
The  land  surface  equals  what  per  cent  of  the  water  surface? 

The  following  table  contains  the  actual  lengths  of  the  coast 
lines  of  the  continent  and  also  the  lengths  that  would  be  needed  to 
enclose  them  if  they  were  solid,  with  smooth  outlines,  also  the 
ratio  of  low  land  to  high  land  : 

ACTUAL  LENGTH,    RATIO  Low  TO 

LENGTHS          nr  SOLID      HIGH  LAND 

North  America  ........  24,040  10,380  6J  :  9 

South  America  ........  13,600  9,030  9£:2f 

Asia  ..................  30,800  13,780  10^:13 

Africa....  ............  14,080  11,760  6J:14| 

Europe  ...............  17,200  0,630  4|:lf 

Australia  ..........  7,600  5,860  3^  :  1 


PERCENTAGE    AND    INTEREST 


253 


11.  Find  what  per  cent  of  each  number  in  column  2  equals  the 
corresponding  number  in  column  3.     What  does  each  per  cent 
mean?     Which  continent  has  the  longest  coast  to  defend  in  com- 
parison with  its  area? 

12.  Express  the  given  ratios  of  column  4  of  low  land  to  high 
land  in  per  cent,  and  tell  what  the  per  cents  mean? 

The  table  below  contains  the  heights,  in  feet,  of  mountains 
and  peaks  of  the  world: 

Popocatapetl  (vole. ) . .  1 7, 784 

Mt  Wrangell 17,500 

Mt.  Blanc 15,744 

Mt.  Shasta 14,350 

Longs  Peak 14,271 

Pikes  Peak 14,147 

Mt.  Etna  (volcano)  ..  10,875 

Rocky  Mts 10,000 

13.  The  height  of  Pikes  Peak  equals  what  per  cent  of  that  of 
Mt.  Everest?  of  Mt.  Logan?  of  Chimborazo?  of  Mt.  Shasta? 

14.  Answer  other  similar  questions  on  the  table. 

15.  What  per  cent  of  North  America  (16,130,269  sq.  mi.)  is 
drained  through  the  Mississippi  basin  (1,250,000  sq.  mi.)?  through 
the   St.  Lawrence  basin   (360,000  sq.  mi.)?  the  Columbia  basin 
(290,000  sq.  mi.)?  the  Colorado  basin  (230,000  sq.  mi.)? 

16.  What  per  cent  of   the  length  of   the    Mississippi   River 
(4,200  mi.)  is  the  length  of  the  St.  Lawrence  (2,000  mi.)?  of  the 
Columbia  River  (1,400  mi.)?  of  the  Colorado  River  (2,000  mi.)? 
Yukon  River  (2,000  mi.)? 

The  following  table  gives  the  lengths,  in  miles,  and  the  areas 
of  basins,  in  square  miles,  of  24  of  the  longest  rivers  of  the  world: 


Mt.  Everest 29,002 

Aconcagua 23,910 

Chimborazo  (volcano)  20,500 

Kilimanjaro  Mts 20,000 

Mt.  Logan 19,500 

Karakoram  Mts 18,500 

Orizaba  (volcano) 18.312 

Mt.  St.  Elias  18,100 


AREA  OF 
RIVER  LENGTH       BASIN 

Mississippi 4,200  1,250,000 

Nile 3,900  1,300,000 

Amazon 3,600  2,500,000 

Yangtzekiang  . .  3,300  650,000 

Obi 3,000  1,000,000 

Yenisei 3,000  1,400,000 

Congo 3,000  1,500,000 

Niger 2,900  1,000,000 

Hoangho 2,800  390,000 

Amur 2,700  780,000 

Lena 2,700  900,000 

La  Plata 2,500  1,350,000 


AREA  OF 

RIVER               LENGTH  BASIN 

Mackenzie 2,400  680,000 

Volga 2,400  590,000 

St.  Lawrence  . .  2,000  360,000 

Yukon 2,000  440,000 

Brahmaputra  ..  2,000  426,000 

Colorado 2,000  230,000 

Indus 2,000  325,000 

Euphrates 2,000  490,000 

Danube 1,900  320,000 

Rio  Grande 1,800  225,000 

Ganges 1,800  450,000 

Orinoco 1,500  400,000 


254  RATIONAL   GRAMMAR   SCHOOL   ARITHMETIC 

17.  What  per  cent  of  the  length  of  the  Mississippi  equals  that 
of  the  Nile?     What  per  cent  of  the  area  of  the  Mississippi  basin 
equals  the  Nile  basin? 

18.  Solve  other  similar  problems  on  the  table. 

The  following  table  contains  the  areas,  in  sq.  mi.,  the  alti- 
tudes, in  ft.  (above  sea  level),  and  the  greatest  depths,  in  ft.,  of 
the  great  lakes  of  the  world  : 

LAKE  AREA  ALTITUDE  DEPTH 

Superior  ..............  31,200  602  1,008 

Huron  ...............  23,800  581  700 

Michigan  .............  22,450  581  875 

Erie  ..................  9,950  573  212 

Ontario  ..............  7,242  248  738 

Victoria  ..............  22,167  4,000  620 

Winnipeg  .............  9,400  710  72 

19.  The  area  of  Lake  Michigan  equals  what  per  cent  of  that  of 
Lake  Superior? 

20.  Similarly  compare  the  altitudes  and  greatest  depths  of 
these  lakes. 

21.  Solve  similar  problems  on  the  table. 

§167.  Commission. 

DEFINITION.—  Commission  is  a  sum  of  money  paid  by  a  person  or  firm 
called  the  principal,  to  an  agent  for  the  transaction  of  business.  It  is 
usually  reckoned  as  some  per  cent  of  the  amount  of  money  received  or 
expended  for  the  principal. 


1.  Find  the  commission  on  $1850  at  2-J-%  ;  at  3-j-%  ;  at 

2.  A  principal  sent  his  agent  $2652  to  be  invested  after  deduct- 
ing the  agent's  commission  of  2  %  •  How  much  money  was  invested? 

QUERIES.  —  Of  what  two  amounts  is  $2652  the  sum?  What 
per  cent  of  itself  is  the  amount  to  be  invested?  What  per  cent  is 
the  commission  of  the  amount  to  be  invested?  What  per  cent 
is  the  total,  $2652,  of  the  amount  to  be  invested? 

SOLUTION.  —  Call  the  unknown  amount  to  be  invested,  x.  Then  1.02  x 
=  $2652. 


3.  5%  commission  on  a  certain  amount  of  money  was  $684.20. 
What  was  the  amount?      (Statement:  .05^  =  $684.20.     Find  x.) 

DEFINITION.  —  A  shipment  of  goods  sent  to  an  agent  to  be  sold  is  called 
a  consignment. 


PERCENTAGE    AND    INTEREST  '255 

4.  A  consignment  of  4560  bu.  of  wheat  was  sold  by  an  agent 
at  78f  ^  per  bushel.     What  was  the  agent's  commission  at  1^  per 
cent? 

5.  A  commission   agent  sold    the    following  consignment  of 
goods:  20  doz.  eggs  at  14f^-;  40  Ib.  creamery   butter  at  210;  36 
Ib.  cheese  at  13J$;  80  Ib.  chickens  at  12-J-r/;   8  doz.  live  chickens 
at  $3. 75;  4  bbl.  apples  at  $3.25;  16  boxes  oranges   at  $2.25;  4 
boxes  oranges  at  $2.50;  12  boxes  graps  fruit  at  $2.75;  25  bunches 
bananas  at  $1.25;  12  boxes  lemons  at  $2.75;  16  bbl.  potatoes  at 
$4.25.     Find  the  agent's  commission  at  6£  per  cent. 

6.  An  agent's  commission   at  2-j-%   on  a    certain    collection 
amounted  to  $97.16.     What  was  the  amount  of  the  collection? 

7.  A  consignment  of  1200  Ib.  cut  loaf  sugar  at  $5.75  per  hun- 
dredweight was  sold  at  12£%  commission.     What  amount  of  money 
was  remitted  to  the  principal? 

8.  A  land  agent  sold  the  N-J-  NW£  section  28  at  $37.50  per  acre. 
His  commission   was  $75.     What  was   the  rate  per  cent  of  his 
commission? 

9.  An  agent  charged  $120  for  selling  a  $3000  piece  of  prop- 
erty.    What  was  the  rate  per  cent  of  his  commission? 

DEFINITION. — The  net  proceeds  of  a  sale  means  the  amount  left  after 
deducting  the  commission  and  other  expenses. 

10.  The  net  proceeds  of  a  sale  of  real  estate  were  $19,400  and 
the  agent's  commission  was  3%.     How  much  did  the  real  estate 
sell  for? 

11.  A  piece  of  land   sold  for  $265.     The  net  proceeds  were 
$240.     What  was  the  rate  per  cent  of  commission? 

§168.  Trade  Discount. 

DEFINITION.  — A  discount  is  a  certain  rate  per  cent  of  reduction  from 
the  listed  prices  of  articles.  The  discount  is  usually  allowed  for  cash  pay- 
ments or  for  payment  within  a  specified  time. 

1.  A  retail  merchant  buys  silk  at  $1.20  a  yard.  He  is  allowed  a 
discount  of  10%  for  cash.  He  pays  cash.  How  much  does  the 
silk  cost  him? 

SOLUTION.  -§1 .20  —  10#  of  §1.20  =  $1.08. 


250  RATIONAL   GRAMMAR   SCHOOL   ARITHMETIC 

2.  A  publisher  sells  50  books  at  $1.50  and  allows  20%  dis- 
count.    How  much  does  he  receive  for  the  books? 

Merchants  and  manufacturers  of  ten  publish  expensive  catalogues 
containing  their  price  lists  of  articles,  whose  prices  fluctuate  rap- 
idly. When  prices  fall,  instead  of  publishing  new  lists,  they  mark  oft' 
an  additional  discount.  For  example,  the  price  of  a  certain  article 
may  be  catalogued  thus:  $60,  discount,  20%,  10%,  5%.  This 
means  a  20%  reduction,  then  a  10%  reduction  on  the  reduced 
price,  and  then  a  5%  reduction  on  the  second  reduced  price.  It 
would  be  computed  thus : 

$60  $48  $43.20 

.20  .10  .05 

$12.00        $4.80  $2.1600 

$60 -$12 -$48.  $48 -$4.80  =  $43. 20.  $43.20 -$2.16  =$41.04. 

NOTE.— Notice  this  is  not  the  same  as  a  discount  of  20%  -f  10%  -f  5% 
(=  35^ ).  Wherein  is  it  different? 

3.  A  New  York  merchant  sells  to  a  customer  goods  marked 
thus:    Price,  $3500;  disc't,  10%   60  da.  and  5%  for  cash.     What 
must  the  customer  pay  to  settle  the  bill  by  cash  payment? 

NOTE.  —The  customer  gets  the  benefit  of  both  discounts. 
What  would  the  customer  have  paid  if  he  had  been  given  a  single 
discount  of  15  per  cent? 

4.  A  bill  of  plumber's  supplies  was  marked  thus: 

Price:  $7.50,  disc't.  20%  and  7%  and  5%. 
How  much  did  the  supplies  really  cost? 

5.  Compute  the  amount  of  money  needed  to  settle  the  follow- 
ing bills  if  paid  in  cash  or  within  the  shortest  time  mentioned: 

(1)  $45;  discount  25%  60  da.  and  10%  5  days. 

(2)  $180;  discount  16f  %  30  da.,  and  10%  10  da.,  and  5%  cash. 

(3)  $1800;  discount  30%  and  10%  and  7%  cash. 

(4)  $54;  discount  12|%,  and  8%,  and  5%  30  days. 

(5)  46  T.  coal  at  $5.75;  discount  10%  and  2%  cash. 

(6)  50  men's  suits  at  $18;  discount  20%  and  5%  10  days. 

(7)  4  gross  tablets  at  50^  per  doz. ;  discount  20%  and  7%  cash. 

6.  To  what  single  rate  of  discount  is  a  discount  of  20%  and 
5%  equivalent? 

SUGGESTION.— Take  a  base  of  $100. 


PERCENTAGE    AND    INTEREST  257 

§169.  Marking  Goods. 

In  marking  his  goods  a  merchant  uses  the  key  word  "harmo- 
nizes." He  writes  the  selliiuj  price  ahove  a  horizontal  line,  and 
the  cost  price  below  it,  using  the  letters  h-a-r-m-o-n-i-z-e-x  in  order 
for  the  digits  1-2-3-4-5-6-7-8-9-0.  For  example  a  book  sells  at 

&2.50  and  costs  $1.75.     The  mark  would  be  =—  r-'  . 

hio 

1.  Using  the  key  "harmonizes,"  interpret  the  following  cost 
marks  and  find  the  per  cent  of  profit  for  each  cost  mark  : 

ns      /A    io       .  .  a  ao        .  mrz     t  .  mao 
(1)  -  —  ;  (2)  -  ;  (3)  -,-r—  ;   (4)  -  —  ;   (5)  -  —  . 

v  '  ma  '  or  '  v  '   /MW  '       '   rs-s-  '   v  '  rrn 

2.  Complete  these  marks  so  that  the  selling  price  may  be  40% 
greater  than  the  cost  price: 


3.  Articles  bearing  the  following  selling  marks  are  marked  to 
sell  at  a  profit  of  33£%  ;  fill  in  the  cost  mark,  using  the  same  key 
as  above  : 


(i)  -;  (^)          (3)        ;  (*)        ;  (5)  -;  (6)       . 

4.  Using  the  same  key,  mark  articles  costing  the  following 
prices  to  sell  at  a  profit  of  37  1%  : 

(1)400;   (2)  720;  (3)  $1.68;  (4)  $2,88;  (5)  $6.40;  (6)  $8.24. 

5.  Supply  complete  cost  marks  for  articles  sold  at  50%  profit 
the  selling  prices  of  which  are  as  follows: 

(1)  900;  (2)  $1.50;   (3)  $2.50;  (4)  $3.66;   (5)  $7.50;   (6)  $8.100 

6.  Solve  problems  4  and  5,  using  the  key  "black  horse." 

7.  Choose  a  key  word  and  with  it  solve  problems  4  and  5. 

8.  A  merchant  wishes  to  mark  his  goods  so  that  he  may  drop 
10%  below  the  marked  price  and  still  make  20%  of  the  cost  price. 
Using  the  key  "harmonizes,"  how  must  he  mark  articles  costing 
the  following  prices  : 

(1)  500?  (2)  600?  (3)  $1.20?  (4)  $2.50?  (5)  $6.40? 


258  RATIONAL   GRAMMAR    SCHOOL    ARITHMETIC 

§170.   Interest.  ORAL  WORK 

Review  §86,  pp.  131-32.  The  method  of  §86  is  known  as  the 
six  per  cent  method. 

In  reckoning  interest  the  year  is  regarded  as  containing  12 
mo.  of  30  da.  each. 

1.  A  man  is  charged  $6  for  the  use  of  $100  for  1  yr.     What 
per  cent  of  the  sum   borrowed    ($100)   equals  the  sum  ($6)   he 
is  charged  for  its  use? 

2.  A  man  is  charged  $21  for  the  use  of  $350  for  1  yr.     The 
sum  charged  equals  what  per  cent  of  the  sum  borrowed? 

DEFINITIONS. — Interest  is  money  charged  for  the  use  of  money.  It  is 
reckoned  at  a  certain  rate  per  cent  of  the  sum  borrowed  for  each  year  it 
is  borrowed. 

When  money  earns  3,  6,  7,  or  10  cents  on  the  dollar  annually  (each 
year)  the  rate  is  said  to  be  3%,  6%,  1%,  or  10%  per  annum  (by  the  year) 
and  the  rate  per  cent  is  said  to  be  3,  6,  7,  or  10. 

3.  At  6%  per  annum,  how  much   interest  does  $360  earn  in 
1  yr.?  in  3  yr.?  in  4|  yr.?  in  2f  yr.?  in  t  years? 

4.  Make  a  rule  for  computing  the  interest  on  any  sum  of 
money  at  6%  when  the  time  is  in  years. 

5.  At   6%   per  annum,   how  much  interest  does  $1  earn  in 
1  yr.?  in  2  mo.?  in  1  mo.?  in  6  da.?  in  12  da.?  in  18  days? 

6.  When  the  time  is  in  months,  how  may  the  interest  on  any 
sum  of  money  at  6%  per  annum  be  computed? 

7.  How  may  the  interest  at  6%  per  annum  on  any  sum  of 
money  be  computed  when  the  time  is  given  in  months  and  days? 
in  years,  months,  and  days? 

DEFINITIONS. — The  sum  of  money  on  which  the  interest  is  computed 
is  called  the  principal.  The  principal  plus  the  interest  is  called  the 
amount.  Since  the  borrower  must  not  only  pay  the  interest  on  the  bor- 
rowed principal,  but  also  return  the  principal,  the  debt  he  must  discharge 
is  the  amount. 

8.  How  long  will  it  require  any  principal  (say  $1)  to  amount 
to  twice  its  value  ($2)  or  to  double  itself  at  6%  per  annum? 

9.  Give  the  reasons  for  these  statements : 

Any  principal  at  6%  —  (1)  doubles  itself  in  200  months;  (2) 
earns  y-J-Q-  of  itself  in  2  mo.  or  60  days. 


PERCENTAGE    AND    INTEREST  259 

10.  Let  /  denote  the  interest  on  $p  at  r%  for  t  yr.  and  let  i 
denote  the  interest  on  %p  at  6%  for  t  yr.  Explain  the  meaning 
of  the  equations  : 

*< 


*•  (f) 


WRITTEN    WORK 

To  find  the  Interest. 

1.  What  is  .07  of  $450?     What  is  the  interest  on  $450  at  7% 
for  1  yr.?  for  2  yr.?  for  5  yr.?  for  3|  yr.?  for  4f  yr.?   for  t  years? 

2.  What  is  .08  of  $1250?     What  is  the  interest  at  8%  on  $1240 
for  1  yr.?  for  3  yr  ?  for  5[f  yr.?  for  x  years? 

3.  $640  was  on  interest  at  5%  from  July  1,  1896,  to  July  1, 
1900.    ^Find  the  interest. 

4.  $1800  was  on  interest  at   7%  for  3  yr.  7  mo.  21  da.     Find 
the  interest. 

CONVENIENT  FORMS 
I.  Interest  computed  first  at  the  given  rate. 

$1800  principal 

.07  rate 

$126.00  int.  for  1  yr. 

3  whole  years 

$378.00  int.  for  3  yr. 

£  of  §126.00     63.00  int.  for  6  mo. 

i  of   63.00     10.50  int.  for  1  mo.  ' 

£of   10.50      5.25  int.  for  15  da. 

I  of   10.50      2.10  int.  for  6  da. 

$458.85  int.  for  6  yr.  7  mo.  21  da. 

II.  Interest  computed  first  at  6%. 

.18  =  int.  on  $1  for  3  yr.  at  6% 
.035  =  int.  on  $1  for  7  mo.  at  6% 
.0035  =  int.  on  $1  for  21  da.  at  6% 


.2185  =  int.  on  $1  for  3  yr.  7  mo.  21  da.  at  6% 
1800 


1748000 
2185 


393.3000  =  int.  on  §51800  for  given  time  at  6% 
65.55      =  int.  on  §1800  for  given  time  at  \% 

$458.85      =  int.  on  $1800  for  given  time  at  1% 

SUGGESTIONS  FOR  II.— 

(1)  If  the  interest  on  $1  for  1  yr.  is  6^,  what  is  the  interest  for  3  years? 

(2)  If  the  interest  on  $1  for  2  mo.  is  IP,  what  is  the  interest  for  7  mo.? 

(3)  If  the  interest  on  $1  for  6  da.  is  1  mill,  what  is  the  interest  for  21  da.  ? 


200  RATIONAL    GRAMMAR   SCHOOL    ARITHMETIC 

5.   How  much  interest  must  I  pay  for  the  use  of  $600  for  1  yr. 

5  mo.  24  da.  at  7  per  cent? 

0.  Find   the   amount    of    $300   for    2    yr.    4   mo.   25   da.   at 
4|  per  cent. 

7.  Find  the  interest  and  the  amount  under  the  following  con- 
ditions : 

(1)  $700  for  1  yr.  7  mo.  15  da.  at  3  per  cent. 

(2)  $400  for  2  yr.  9  mo.  27  da.  at  3  per  cent. 

(3)  $210  for  2  yr.  5  mo.  28  da.  at  6  per  cent. 

(4)  $150  for  1  yr.  11  mo.  13  da.  at  7  per  cent. 

(5)  $280  for  1  yr.  G  mo.  19  da.  at  4|  per  cent. 
(G)  $3GO  for  1  yr.  4  mo.  5  da.  at  4  per  cent. 

(7)  $260  for  2  yr.  3  mo.  11  da.  at  3£  per  cent. 

(8)  $500  for  1  yr.  11  mo.  14  da.  at  7  per  cent. 

(9)  $300  for  2  yr.  7  mo.  12  da.  at  5  per  cent. 
(10)  $625  for  3  yr.  9  mo.  18  da.  at  6  per  cent. 

To  find  the  Principal. 

8.  What    principal    at    8%    will    furnish   $16   interest   in   2 

years? 

• 

SUGGESTION. —What  interest  will  $1  produce  at  8%  in  2  yr.?    How 
many  dollars  will  yield  $16  at  8%  in  2  years? 

9.  What   principal   at   8%   will   produce   $30   interest  in   2£ 
years? 

10.  What  principal  at   6%  will  amount  to  $112  in   1  yr.    6 
months? 

SUGGESTION.— What  is  the  amount  of  $1  at  6%  interest  for  1  yr.  6  mo.? 

11.  Find  the  principal  which  will  yield  $61.25  interest  in  3  yr. 

6  mo.  at  7  per  cent. 

12.  Find  the  principal  which  will  amount  to  $972.40  in  3  yr. 
2  mo.  12  da.  at  4|  per  cent. 

13.  Make  a  rule  for  finding  the  principal  when  the  rate,  time, 
and  interest  are  given. 

14.  Make  a  rule  for  finding  the  principal  when  the  rate,  time, 
and  amount  are  given. 


PERCENTAGE    AND    INTEREST  261 

15.  Supply  the  correct  value  for  the  letter  in  each  of  the  fol- 
lowing cases: 

RATE                      TIME                             INTEREST      PRINCIPAL  AMOUNT 

(1)  6   %          1  yr.  7  mo.  15  da.           $19.50             P  A 

(2)  8   %         3  yr.  3  mo.  18  da.         $169.02             P  A 

(3)  8£%         4  yr.  7  mo.  21  da.              /                  P  .$998.50 

(4)  7   %          6  yr.  8  mo.  16  da.               /                   P  $198.42 

(5)  4   %          2  yr.  6  mo.  18  da.         $122.50             P  A 

6)  5  %         1  yr.  10  da.                    $176.00            P  A 

7)  4i%         90  da.                                $32.50             P  A 


To  find  the  Rate. 

16.  At  what  rate  per  cent  will  $320  yield  $34  interest  in  2  yr. 
9  months? 

SUGGESTION.—  How  much  interest  will  $320  yield  in  2  yr.  9  mo.  at  \%1 
At  what  rate  per  cent  then  will  the  same  sum  yield  834  interest  in  2  yr. 
9  months? 

17.  At  what  rate  will  $780  yield  $486.60  in  5  yr.  8  months? 

18.  A  man  invested  $2000  for  2  yr.  7  mo.  27  da.  and  received 
$2638  at  the  end  of  this  time.     What  rate  per  cent  of  interest 
did  his  investment  earn  for  him? 

19.  A  man  bought   120  A.  of  land  at  $85  and  sold  it  2  yr. 
8  mo.  later  for  $100  per  A.,  after  having  received  $900  in  rents 
from  it  and  having  twice  paid  taxes  on  it  at  75  cents  per  acre. 
What  was  his  annual  rate  per  cent  of  profit? 

20.  Make  a  rule  for  finding  the  rate  when  the  principal,  time, 
and  interest  are  known. 

21.  Supply  the  correct  value  to  the  first  decimal  place  for  the 
letter  in  each  of  the  following  problems  : 

(1)  $580.00  1  yr.  5  mo.  $46.50         r% 

(2)  $1280.00  3  yr.  10  mo.         $500.00          r% 

(3)  $798.45  2  yr.  8  mo.  15  da.         $258.65          r% 

(4)  $3698.50  1  yr.  5  mo.  19  da.         $568.75          r% 

To  find  the  Time. 

22.  How   long  will  it  take  $80  to  earn  $14  interest  at  4% 
annually? 

SUGGESTION.  —  How  much  interest  will  $80  earn  in  one  year  at  4%  ?  In 
how  many  years  then  will  $80  earn  $14  at  the  same  rate? 


262  RATIONAL   GRAMMAR    SCHOOL   ARITHMETIC 

23.  How  long  will  it  take  $125  to  earn  $57.50  interest  at  8% 
per  annum? 

24.  At  7%,  how  long  will  it  take  $648  to  yield  $69.84? 

NOTE. — When  the  time  results  in  decimals  of  a  year  the  decimal  may 
be  reduced  to  months  and  days  by  the  method  of  problem  65,  p.  235. 

25.  How  long  will  it  take  $750  to  yield  $750  interest  at  8%? 

26.  How  long  will  it  take  $10  to  double  itself  at  6%?  at  7%? 

27.  How  long  will  it  take  $975  to  amount  to  $1225  at  5%? 

SUGGESTION. — What  is  the  total  interest?     What  is  the  interest  on 
$975  at  5%  for  1  year? 

28.  Make  a  rule  for  finding  the  time,  T,  required  for  a  given 
principal,  P,  to  amount  to  a  given  sum,  A,  at  a  given  rate  per 
cent,  r? 

29.  Make  a  rule  for  finding  how  long  it  will  take  a  given 
principal  to  earn  a  given  interest  at  a  given  rate  per  cent. 

30.  Find  the  time,  T,  in  years,  months  and  days  under  the 
conditions  stated  in  each  of  the  folloAving  problems : 

PRINCIPAL  RATE  INTEREST  TIME 

(1)  $66.00  7   %  $28.60  T 

(2)  $460.00  6   %  $31.05  T 

(3)  $750.00  5i%  $147.475  T 
(4). $1260.00  5   %  $213.15  T 
(5)  $2460.00  4  %  $321.44  T 

31.  Supply  the  value  for  which  each  letter  stands  in  the   prob- 
lems of  the  following  table: 

PRINCIPAL     RATE  TIME  INTEREST  AMOUNT 

(1)  $60.00        6%         3  yr.  3  mo.    8  da.  I  A 

(2)  $175.00        5%         4  yr.  9  mo.  15  da.  I  A 

(3)  $800.00        1%  T  I  $926.00 

(4)  $475.00        8%  T  $142.50  A 

(5)  $1266.00        \%  ,    T  I  $1349.85 

(6)  P  6%         5  yr.  7  mo.  27  da.         8509.25  A 

(7)  $1575.30        r%         1  yr.  4  mo.  18  da.  /  $1662.46| 

(8)  $728.25        Sft  T  $209.736  A 

(9)  $364.75       T%        2  yr.  1  mo.  15  da.          $69.29  A 


PERCENTAGE   AND   INTEREST 

32.  Complete,  to  mills,  the  following  interest  table : 
INTEREST  TABLE:    PRINCIPAL  $100. 


263 


RATE 

8# 

4# 

b% 

8# 

7# 

1  da  

$.008 

$.011 

§.014 

$.017 

$.019 

2  da  

3  da  

4  da 

5  da  

6  da 

1  mo  

2  mo  

3  mo 

Q  nio  

1  VI* 

33.  Compute  by  the  table  the  interest  on  $758  at  7%   for 
4  mo.  12  da. ;  for  7  mo.  8  da. ;  for  1  yr.  3  mo.  10  days. 

§171.  Algebra. 

1.  Compute   the   interest   on  $750   at   5%   for   each   of  the 
following  times:     (1)   1  yr. ;    (2)  2  yr. ;    (3)  6^  yr. ;   (4)  3f  yr. ; 
(5)  25|  F. ;  (6)  x  yr. ;  (7)  t  years. 

2.  Find  the  interest  and  the  amount  on  $1250  for  each  of 
the  following : 


(1)  5%,  1    yr. 

(2)  6%,2fyr. 

(3)  7%,  6    yr. 


(4)  6%,  3£yr. 

(5)  4%,  7fyr. 

(6)  3%,  t    yr. 


(8) 
(9) 


3.  Denote  the  interest  on  a  certain  principal,  P,  by  /,  the  rate 
by  r,  and  the  time  (in  years)  by  £,  write  an  equation  showing  how 
to  find  /  from  P,  r,  and  t,  and  translate  into  words  the  meaning 
of  the  equations. 


264  RATIONAL    GRAMMAR    SCHOOL   ARITHMETIC 

SOLUTION.— 


(I)  -I 


(1)  J^PX—xe. 


Translated  into  words:  (1)  means,  "Interest  equals  the  product  of  the 
principal,  the  rate  divided  by  100,  and  the  time  (in  years).  '; 

(2)  and  (3)  mean,  "Interest  equals  the  product  of  principal,  rate,  and 
time,  divided  by  100." 

4.  Calling  A  the  amount,  /  the  interest,  and  P  the  principal, 
write  an  equation  showing  how  to  find  A  from  /  and  P. 

5.  Write  an  equation  to  show  how  to  find  P  from  /,  r,  and  t, 
and  state  in  words  the  meaning  of  the  equation. 

If    we    multiply   both  sides    of  the  equation,   I  =   --^r,  by  100  we 

1UU 

have  the  equation,  100  /=  Prt.  Now  to  show  how  to  find  P  from  7,  r,  and 
t,  divide  both  sides  by  rt,  and  write  the  second  member  on  the  left.  We 
then  have: 


6.  State  in  words  the  meaning  of  formula  (II). 

7.  Show,  hy  proper  multiplications  and  divisions  of  equation 
(I)  (3),  that  the  rate  r  may  be  found  from  P,  /,  and  ^,  by  the 
equation, 

(HI)         r  =  ^. 

8.  State  in  words  the  meaning  of  formula  (III). 

9.  Show  from  (I)  (3)  that  the  time  t  may  be  found  from  /,  P, 
and  r  by  the  formula, 


10.  State  in  words  the  meaning  of  formula  (IV). 

11.  Solve  by  formulas  I-IV  the  following  problems: 

(1)  P  =  $64,  r  =  8,  and  t  =  2±;  find  /. 

(2)  /=  $24,  r  =  6,  and  t  =  H;  find  P. 

(3)  /=  $75,  P  =  $850,  and  t  =  2;  find  r. 

(4)  /  =  $120,  P  =  $600,  and  r  =  10;  find  t. 

(5)  /=  $230.13|,  P  =  $722,  and  t  =  3f;  find  r. 


PERCENTAGE   AND   INTEREST  265 

§172.  Promissory  Notes. 

DEFINITIONS.  —  A  promissory  note  is  a  written  promise,  made  by  one 
person  or  party,  called  the  maker,  to  pay  another  person  or  party,  called 
the  payee,  a  specified  sum  of  money  at  a  stated  time. 

The  sum  of  money  for  which  the  note  is  drawn  is  called  the  face  value, 
or  the  face,  of  the  note. 

The  date  on  which  the  note  falls  due  is  called  the  date  of  maturity, 
and  the  time  to  run  is  the  time  yet  to  elapse  before  the  note  falls  due. 

/?J^r  KANKAKEE,  ILL.,    fane  /,  S903. 

*$mefu  dm/4  "after  date  I  promise  to  pay  to  the 

order  of    S&.   ^M.    98a4e9*f    Jvo    tTCt&ubeiJ  ^7i/£/   and  wo 

*  f 

Dollars,  for   value   received,  with   interest   at   fix  per  cent  per 
annum  from  date. 

.    /,   S903.  2^3$.  Jrewman. 


1.  Who  is  the  maker  of  the  above  note?  the  payee?    What  is 
the  face  of  the  note?  the  date?  the  rate  of  interest?  the  date  of 
maturity?  the  time  to  run? 

2.  A  promissory  note,  unless  otherwise  specified  in  the  note, 
draws  interest  on  its  face  value  at  the  rate  mentioned  in  the  note 
from  the  date  of  the  note  until  it  is  paid.     Compute  the  interest 
and  the  amount  on  the  foregoing  note  if  it  was  paid  Sept.  1,  1903. 

3.  Find  the  interest  on  the  following  note,  paid  Feb.  10,  1903: 

$875.  Urbana,  111.,  May  18,  1897. 

One  year  after  date  I  promise  to  pay  to  James  Black,  or 
order,  Eight  Hundred  Seventy-five  and  ^  Dollars,  at 
Busey's  Bank,  for  value  received,  with  interest  at  the  rate 
of  seven  per  cent  per  annum  from  date. 

Due  May  18,  1898.  HENRY  OSBORN. 

NOTE.  —  To  find  the  time  for  which  interest  is  to  be  computed,  pro- 
ceed thus  : 

CONVENIENT  FORM  EXPLANATION.  — 

Date  of  payment,  1903     2     10  2  mo.  10  da.  =  1  mo.  40  da. 

Date  of  note  1897     5     18  *  mo-  40  da-  —  18  da-  =  l  mo-  23  da- 

1903  1  mo.  =  1902  13  mo. 


.  . 

Time,  5     8     22  !902  13  mo.  —  5  mo.  =  1902  8  mo. 

yr.  mo.  da.  1902  —  1897  =  5  yr. 


266  KATIOtfAL   GRAMMAR   SCHOOL   ARITHMETIC 

4.  Find  the  amount  of  each  of  the  following  notes : 


PACE 
(1)  $250 

RATE 
6  % 

DATE  OF  NOTE 
Mar.  12,  1899 

DATE  OF  PAYMENT 
Jan.  1,  1902 

INTEREST  AM'NT 

(2)  $635 

7  % 

Nov.  20,  1896 

July  15  1900 

(3)  $2400 

5±% 

Dec.  12,  1899 

Jan.  8,  1903 

(4)  $3865 

4i% 

Oct.  13,  1901 

Sept.  7,  1903 

(5)  $3640 

8*<& 

Aue.  17,  1900 

Mar.  3,  1903 

§173.  Discounting  Notes. 

DEFINITION.  —  Discount  is.  a  deduction  from  the  amount  due  on  a  note 
at  the  date  of  maturity. 

In  some  cases  promissory  notes  do  not  draw  interest.  The  fol- 
lowing is  an  example: 

John  C.  Cannon  purchased  a  self  -binding  harvester  from  A.  R. 
Crow  for  $120  and  gave  him  the  following  note  in  payment: 


Peoria,  111.,  June  20,  1899. 

Eighteen  months  after  date,  for  value  received,  I  promise 
to  pay  to  A.  R.  Crow,  or  order,  One  Hundred  Twenty  and 
ffo  Dollars,  without  interest  until  due. 

Due  Dec.  20,  1900.  JOHN  C.  CANNON. 

1.  On  Sept.  20,  1899,  Crow  sold  the  note  to  Adams  at  such  a 
price  that  Adams  received  his  purchase  money  and  8%  interest  on 
it  until  the  date  of  maturity   (Dec.  20,  1900).     How  much  did 
Adams  pay  for  the  note? 

SUGGESTIONS.—  Any  principal  at  8%  will  amount  to  110%  of  itself  in 
1  yr.  3  mo.  Why?  Hence,  §120  =  110^  of  what  number?  Or,  better, 
l.lOa?  =  $120.  Find  the  value  of  x. 

DEFINITIONS.  —  The  sum  of  money  which,  at  the  specified  rate  and  in 
the  time  the  note  is  to  run  before  falling  due,  will  amount  to  the  value  of 
the  note,  when  due,  is  called  the  present  worth  of  the  note.  The  difference 
between  the  value  of  the  note,  when  due,  and  the  present  worth  is  called 
the  true  discount. 

The  bank  discount  of  a  note  is  the  interest  upon  the  value  of  the  note 
when  due,  from  the  date  of  discount  until  the  date  of  maturity. 

2.  If  Crow  had  sold  the  above  note  to  a  banker  at  a  discount 
of  8%,  the  banker  would  have  computed  the  interest  at  8%  on 
1120  from  the  date  of  sale  (Sept.  20,  1899)  until  the  date  of 
maturity.     How   much   would    he   have   received   for  the  note? 
Find  the  difference  between  the  true  and  the  bank  discount  of 
the  note. 


PERCENTAGE  AND  INTEREST 


267 


3.  C.  A.   Thomas  bought  a  road  wagon  of  J.  K.  Duncan, 
giving  the  following  note  in  payment : 

$65.  Pekin,  111.,  Sept.  10,  1900. 

Two  years  after  date  I  promise  to  pay  J.  K.  Duncan,  or 
order,  Sixty-five  Dollars,  value  received,  with  interest  at  1% 
per  annum.  C.  A.  THOMAS. 

Due  Sept.  10,  1902. 

On  March  10,  1901,   Duncan  sold  this  note  to  a  bank  at  7£% 
discount.     How  much  did  Duncan  receive  for  the  note? 

NOTE. — Remember,  bank  discount  is  computed  on  the  amount  of  the 
note  when  due,  for  the  time  to  run  from  date  of  sale. 

4.  Counting  money  worth  7%,  how  much  did  Duncan  receive 
for  the  wagon? 

5.  A  man  bought  a  horse,  giving  in  payment  his  note  for 
1  yr.  for  $85,  dated  Feb.  26,  1903,  and  drawing  interest  at  1% 
from  date.     Two  months  later  the  holder  of  the  note  discounted 
it  at  a  bank  at  6  % .     What  was  the  discount?    What  did  the  bank 
pay  for  the  note?  v 

6.  Find   the  bank   discount  and  proceeds  on  the  following 
notes : 


FACE    RATE     DATE  OF  NOTE 


DATE  or 
MATURITY 


DATE  OF 

SALE 


(1)  $60  6 

(2)  $275 

(3)  $350 

(4)  $700  8 

(5)  $858  6 

(6)  $1260  5 

(7)  $1800 

(8)  $2450  7 

(9)  $3865  6 


Apr.     6,  1897  Oct.     6,  1898  June  15,  1897 

Aug.    7,  1899  Nov.    7,  1901  ,  Mar.  13,  1900  6  '% 

Sept.  10,  1900  Dec.  10,  1902  Jan.   15,  1901  6  % 

Feb.   15,  1901  Nov.  15,  1903  June  20,  1901  6  % 

Jan.     1,  1900  July  15,  1903  May  19,  1900  7  % 

Feb.  28,1896  Aug.  31,  1900  Aug.    8,1896  6  % 

June  19,  1897  Aug.  19,  1901  Dec.  29,  1898  6  % 

Nov.  18,  1899  Feb.  28,  1902  Dec.     1,  1900  fy% 

Dec.  20,  1901  Nov.  20,  1902  Feb.  12,  1902  5  % 


RATE    BANK 

OF       Dis- 

Disc.  COUNT 

7  %    . 


(10)18600    4|£     Oct.    18,1900    Jan.  18,1904    Apr.    2,1901    ±%    

§174.  Partial  Payments. 

DEFINITION. — When  a  note  or  bond  is  paid  in  part  the  fact  is 
acknowledged  by  the  holder  by  his  writing  the  date  of  payment,  the 
sum  paid,  and  his  signature  on  the  back  of  the  note  or  bond.  This  is 
called  an  indorsement. 

Partial  payments  are  made  only  (1)  on  notes  which  read,  "On  or 
before,  etc.,"  (2)  by  private  agreement  between  the  maker  and  the  holder 
of  the  note. 


268  RATIONAL    GRAMMAR    SCHOOL   ARITHMETIC 

For  calculating  the  balance  due  on  a  note  or  bond  on  which 
partial  payments  have  been  made,  nearly  all  the  states  have 
adopted  the  following  rule,  known  as  "The  United  States  Kule 
of  Partial  Payments,"  which  has  been  made  the  legal  rule  by  a 
decision  of  the  Supreme  Court  of  the  United  States: 


RULE. — Find  the  amount  of  the  principal  to  the  time 
the  payment  or  the  sum  of  the  payments  equals  or  exceeds  the 
interest  due;  subtract  from  this  amount  the  payment  or  the  sum 
of  the  payments.  Treat  the  remainder  as  a  new  principal  and 
proceed  as  before, 

ILLUSTRATIVE    EXAMPLES 

1.  Find  the  balance  due  on  the  following  note  at  maturity: 

$1250.  Chicago,  111.,  May  21,  1900. 

On  or  before  two  years  after  date  I  promise  to  pay  to  the 
Order  of  P.  A.  Hopper  Twelve  Hundred  Fifty  and  T°0°o 
Dollars  at  the  Corn  Exchange  National  Bank,  for  value 
received,  with  interest  at  6  per  cent  per  annum. 

Due  May  21,  1902.  JOHN  P.  MILLER. 

The  indorsements  on  the  back  of  this  note  were  as  follows : 
Nov.  21,  1900  . .  $80.00 


Feb.  21,  1901  ...................     $10.00 


May  21,  1901  ...................  $150.00 


Feb.  21,  1902  ...................  $500.00 


PERCENTAGE    AND    INTEREST  269 

SOLUTION  BY  RULE 

Principal  on  May  21,  1900 $1250.00 

Interest  for  6  mo.  on  §1 .03 


Interest  due  Nov.  21,  1900 I    37.50 


Amount  Nov.  21,  1900  (date  of  first  payment  of  §80) $1287.50 

First  payment 80.00 

New  principal  Nov.  21,  1900 $1207.50 

Interest  for  3  mo.  on  $1 .015 


Interest  to  Feb.  21,  1901  (date  of  second  payment  of  $10)  ....     $    18.11 
Payment  being  less  than  interest  no  settlement  is  made. 

$1207.50 
Interest  for  6  mo.  on  $1 .03 


Interest  to  May  21,  1901  (date  of  third  payment  of  $150) $    36.23 

Amount  May  21,  1901 $1243.73 

Sum  of  second  and  third  payments  ($10  +  $150)  160.00 

New  principal  May  21,  1901 4 $1083.73 

Interest  for  9  mo.  on  $1 _      .045 

Interest  to  Feb.  21,  1902  (date  of  fourth  payment  of  $500)  . . .  $    48.77 

Amount  Feb.  21,  1902 $1132.50 

Fourth  payment 500.00 

New  principal  Feb.  21,  1902 $  632.50 

Interest  for  3  mo.,  on  $1 . .  .015 


Interest  to  date  of  maturity,  May  21,  1902 $      9.49 

Balance  due  at  maturity,  May  21,  1902 $  641.99 

2.  The  following  payments  were  made  on  a  $650  note,  bearing 
7%  interest  and  dated  April  20,  1901: 

July  30,  1901 $75.00 

Jan.    15,  1902 $15.00 

Aug.  12,  1902 $175.00 

Jan.      1,  1903 $50.00 

Find  the  amount  due  April  20,  1903. 

3.  A  note  of  $2800,  dated  Feb.  23,  1900,  and  bearing  7% 
interest,  carried  the  following  indorsements : 

Feb.   23,  1901 $100.00 

July  16,  1901 §50.00 

Jan.      1,  1902 $800.00 

July  15,  1902 $85.00 

Nov.  28,  1902 $380.00 

Find  the  amount  due  Feb.  23,  1903. 


APPLICATIONS  TO  TRANSPORTATION  PROBLEMS 


FIGURE  138 


§175.  Locomotive  Engine. 


1.  The  small  wheels  under  the  front  of  the  engine  are  called 
pilot     wheels,    or  leaders.     How  many  leaders  are  there  under 
the  engine  (on  both  sides)? 

The  large  wheels  are  called  drivers.     The  smaller  wheels  just 
behind  the  drivers  are  called  trailers. 

Answer  the  following  questions  by  referring  to  Fig.  138: 

2.  How  high  is  the  center  line  of  the  boiler  above  the  top  sur- 
face of  the  track? 

3.  Give  the  following  distances  in  feet: 

(1)  Between  the  centers  of  the  leaders; 

(2)  Between  the  centers  of  the  rear  leader  and  of  the  front 
driver ; 

(3)  Between  the  centers  of  the  drivers ; 

(4)  Between  the  centers  of  the  rear  driver  and  of  the  trailer ; 

(5)  Between  the  centers  of  the  trailer  and  of  the  front  wheel 
of  the  tender; 

(6)  Between  the  centers  of  the  front  two  wheels  of  the  tender; 

(7)  Between  the  centers  of  the  front  leader  and  of  the  rear 
tender  wheel ; 

(8)  Between  the  centers  of  the  front  leader  and  of  the  trailer. 

270 


APPLICATIONS   TO   TRANSPORTATION    PROBLEMS  271 

4.  Give  the  distance  in  inches  (1)  between  the  nearest  points 
on  the  rims  of  the  drivers ;  (2)  between  the  nearest  points  on  the 
rims  of  the  leaders;  (3)  between  the  nearest  points  on  the  rims 
of  the  front  tender  wheels. 

5.  How  long  are  the  radii  of  the  leaders  shown  in  the  cut? 
How  long  are  the  circumferences  of  these  wheels? 

6.  Compute  the  radii  and  the  circumferences  of  the  tender 
wheels. 

7.  Find  the  circumference  of  the  drivers ;  of  the  trailers. 

8.  When  the  drivers  turn  over  240  times  a  minute,  how  fast 
does  the  engine  go? 

9.  How  many  times  do  the   leaders  turn  while   the  drivers 
turn  round  once? 

10.  The  weights  written    beneath    indicate    the  number  of 
pounds  of  the  weight  of  the  engine  which  is  borne  by  the  different 
pairs  of  wheels.     Find  the  total  weight  of  the  engine. 

11.  The  weight  of  the  empty  tender  is  43,000  Ib.     When  the 
bender  is  loaded  it  carries  10  T.  coal  and  7000  gal.  of  water.     A 
cubic  foot  of  water  weighs  62-J  Ib.,  and  contains  7-J-  gal.    Find  the 
total  weight  of  the  loaded  tender. 

12.  The  engine  shown  in  the  cut  drew  a  train  of  3  sleepers, 
averaging  93,000  Ib. ;  5  passenger  coaches,  averaging  78,000  Ib. ; 
an  express  car,  weighing  34,000  Ib. ;    and  a  mail  car,  weighing 
75,000  Ib.     What  was  the  total  weight  of   the  train,  including 
both  engine  and  tender? 

13.  The  force  exerted  by  an  engine  to  draw  a  train  is  dif- 
ferent for  different   speeds.     For  a  speed  of    10   mi.  an    hour 
it  has  been  found  that  on  straight,  level  track  an  engine  must 
exert   a  force   of   4f  Ib.    for  each  ton  of  weight  of  the  train, 
including  the  weight   of   both    engine  and   tender.      Find  the 
force   required   to   draw  the  train  of  problem   12   under  these 
conditions. 

14.  For  a  speed  of  15  mi.  an  hour  a  force  of  5£  Ib.  per  ton 
of  train  weight  is  needed.     What  force  will  draw  the  train  of 
problem  12  at  this  speed? 

15.  Following  are  the  forces  for  different  speeds  from  20  to  75 
mi.  an  hour.     Find  the  pulling  (tractive)  force  to  be  exerted  by 


RATIONAL    GRAMMAR   SCHOOL   ARITHMETIC 


the  engine  to  draw  the  train  of  problem  12  at  each  indicated 
speed : 


SPEED 

FORCE  IN  LB. 

PERT. 

TRACTIVE 
FORCE 

SPEED 

FORCE  IN  LB. 

PER  T. 

TRACTIVE 
FORCE 

20 

6i 

50 

ui 

25 

n 

55 

124 

30 

8 

60 

13 

35 

8| 

65 

181 

40 

»l 

70 

14} 

45 

10£ 

75 

15] 

16.  Find  the  difference  between  each  number  and  the  number 

next  above  it  in  the  table.     What  do  you  find? 

y 

17.  In  the  equation  F=  —  +  3,  let  F  stand  for  the  force  in 

pounds  per  ton  and  let  V  stand  for  the  speed  in  miles  per  hour. 
Let  V  =  20  in  the  equation.  Find  F  by  dividing  20  by  6  and 
adding  3  to  the  quotient.  Compare  your  result  with  the  number 
of  column  2,  and  in  line  with  20.  What  do  you  find? 

18.  Let  V  =  25.     Find  F  and  compare  with  the  number  of  the 
table  in  line  with  25. 

19.  Let  V  equal  other  numbers  of  columns  1  or  4  and  find  F 
for  each  speed. 

DEFINITIONS. — Replacing  V  in  this  way  by  numbers  like  20,  25,  and 
so  on  is  called  substituting  for  V  the  numbers  20,  25,  and  so  on. 

Performing  the  operations  indicated  in  the  equation  and  obtaining 
the  number  for  F  is  called  find- 
ing the  value  of  F. 

20.  How  does  the  equa- 
tion say  that  the  force  in  Ib. 

per  ton  (F)  needed  to  draw      '%%  J4? 

the  train  can  be  obtained 
when  the  speed  (F)  of  the 
train  is  known? 

21.  The    speeds    of     the 
table  (problem  15)  are  plotted 
to  scale  on  the  horizontal  line 

of  Fig.  139,  and  the  forces,  in  Ib.  per  ton  of  load,  are  plotted  to  a 
different  scale  on  the  vertical  parallels.  The  points  1  to  12  are 


30  25  30  35  T40  45  50  55  60  65  70  75  80  85  90 

Speed  (V)  in  mi.perhr.  Scale/a"«5mi.perhr. 
FIGURE  139 


APPLICATIONS   TO    TRANSPORTATION    PROBLEMS  273 

the  upper  ends  of  the  vertical  lines,  whose  lengths  represent  the 
successive  numbers  of  columns  2  and  5  of  the  table.  Place  the 
edge  of  a  ruler  along  these  points,  or  stretch  a  thread  taut  just 
over  them.  How  do  the  points  seem  to  lie? 

22.  Notice  the  horizontal  and  the  vertical  scales  and  make  a 

drawing  like  that  of  Fig.  139  to  a  scale  4  times  as  large. 

y 

23.  Assuming  that  the  same  law,  F=  —  +  3,  relating  force  and 

speed,  holds  also  for  a  speed  of  80  mi.  per  hour,  substitute  V-  80 
in  the  equation  and  compute  F,  the  force  in  pounds  per  tonneeded 
to  draw  the  train  80  mi.  per  hour.  Make  a  similar  computation 
for  F=  85  and  for  F=90. 

24.  Plot  to  scale  on  the '80,  85,  and  90  lines  of  your  enlarged 
drawing  the  computed  values  of  F,  these  lines  becoming  13, 14  and 
15.  Be  careful  to  get  each  computed,  F,  on  the  proper  vertical  line. 

25.  Stretch  a  string,  or  place  a  ruler,  along  the  points  1  to  15 
of  your  drawing.      Do  the  points  added  from  your  computed 
values  seem  to  lie  on  the  straight  line  through  the  points  1  to  12? 
With  a  ruler  draw  a  single  straight  line  through  all  the  points  of 
your  drawing. 

26.  Mark  a  point  on  the  horizontal  line  midway  between  the 
35  and  40  points.     What  speed  does  this  point  represent?     Draw 
a  vertical  from  this  point  up  to  the  line  through  the  points  1  to 
12.     Mark   the  upper  end   of  this  vertical   a.     What  force  in 
pounds  per  ton  does  the  line  from  a  to  37|  represent? 

27.  How  could  you  find  from  the  drawing  the    number   of 
pounds   per   ton  needed   to  draw   the  train  42J  mi.  per  hour? 
67£  mi.  per  hour?   22-J-  mi.  per  hour?    15  mi.  per  hour?    17£  mi. 
per  hour?  10  mi.  per  hour?    5  mi.  per  hour? 

28.  How  could  you  find  from  the  equation,  F  =  —  4-  3,  the 

numbers  of  pounds  per  ton  needed  to  draw  the  train  at  the  speeds 
18,  36,  42,  57,  and  69  mi.  per  hour?  Compute  these  forces  and 
compare  them  with  the  numbers  found  from  measurements  on  the 
drawing  of  Fig.  139. 

REMARK.— The  straight  line  through  the  points  1  to  12  is  said  to 
represent  the  equation,  F=  —  -f-  3. 


274  RATIONAL   GRAMMAR   SCHOOL   ARITHMETIC 

§176.  Laws  of  Tractive  Force. 

1.  The  force  (F),  in  pounds,  needed  to  draw  a  load  (L),  in 
pounds,  on  a  common  road  wagon  over  loose  sand  is  given  by 
4F  =  L.     Find  the  forces  (F)  needed  to  draw  these  loads: 

(1)  L  =  864  Ib. ;  (2)  L  =  1280  Ib. ;  (3)  L  =  2648  Ib. ;  (4)  L  = 
3268  Ib. ;  (5)  L  =  4893  Ib. 

2.  Over  fresh  earth,  the  law  is  F=  .125  L.     What  forces  (F, 
in  pounds)  are  needed  to  draw  the  following  loads,  in  pounds, 
over  fresh  earth : 

(1)  L  =  680?  (2)  L  =  1624?  (3)  L  =  2160?  (4)  L  =  3840?  (5)  L 
=  4580? 

3.  With  common  road  vehicles  on  dry  level  highways  the  law 
of  pulling  (tractive)  force  (in  pounds)  is  F  =  .025  L.     L  denotes 
the  combined  weight  of  the  wagon  and  load  in  pounds.     What 
forces  will  be  needed  to  draw  the  following  loads : 

(1)  45  bu.  wheat  on  a  1200-lb.  wagon?  (2)  2  T.  coal  on  a  2200- 
Ib.  wagon?  (3)  1-J-  T.  hay  on  a  1680-lb.  wagon?  (4)  A  traction 
engine  weighing  8^  T.?  (5)  A  thresher,  weighing  4J  T. 

4.  On  well  packed  gravel  roads  the  law  is  F  =  .052  L.     To 
draw  a  certain  load  on  gravel  road  a  tractive  force  of  44.72  Ib.  was 
exerted.     What  was  the  load  L? 

SOLUTION. — Dividing  both  sides  of  the  equation  by  .052,  we  have 

77T  771 

-Trr^  =  L,  or,  what  is  the  same  thing,  L  =  -7^.     We  have  then,  by  sub- 
.Uo/e  .uo^ 

A  A  70 

stituting,  L  =  =%£-  =  860.     Ans.  L  =  860  Ib. 

.UO/3 

5.  Under  the  conditions  expressed  in  problem  4,  find  the  loads 
the  following  forces  will  draw: 

(1)  ^T=54.08lb.;  (2)  ^=66.56  Ib. ;  (3)^=137.8  Ib. ;  (4) 
201.76  Ib.;  (5)  234  pounds. 

6.  The  law  of  tractive  force,  on  good,  straight,  level  railroad 
track  is  F=  .0035  L\  on  fair  track,  straight  and  level,  it  is  F  = 
.0059  L.    Weights  of  cars  are  given  in  problems  15  and  17,  p.  135, 
and  in  problem  12,  p.  271.     Make  and  solve  problems  based  on 
these  facts. 


APPLICATIONS   TO   TRANSPORTATION   PROBLEMS  275 

7.  The  tractive  force  (F)  that  can  be  exerted  by  a  locomotive 
on  dry  track,  straight  and  level,  is  about  .3  of  the  part  of  the 
weight  of  the  locomotive  which  rests  on  the  drivers.     The  loads 
that  can  just  be  moved  along  by  a  locomotive  are  given  by  the 
equations  of  the  last  problem.     Make  and  solve  problems  based 
on  the  following  actual  weights  upon  the  drivers  of  certain  loco- 
motives : 

(1)  80,890  Ib. ;  (2)  85,850  Ib. ;  (3)  86,030  Ib. ;  (4)  106,875  Ib. ; 
(5)  112,190  lb.;  (6)  131,225  Ib. ;  (7)  141,320  Ib.;  (8)  202,232  Ib. 
(See  also  Fig.  139). 

8.  Problems  may  also  be  made  on  the  following  laws  for  road 
wagons : 

(1)  Broken  stone  (fair)      .     .  F=.Q%SL; 

(2)  Broken  stone  (good)    .     .  F=  .015  L\ 

(3)  Worn  macadam  .     .     .     .  F  =  .033  L\ 

(4)  Nicholson  pavement    .     .  F=  .019  L\ 

(5)  Asphalt  pavement  .     .     .  F-  .012  L\ 

(6)  Stone  pavement      ...  ^=.019Z; 

(7)  Granite  pavement  .     .     .  F=  .008  L\ 

(8)  Plank  road F=  .010  L. 

9.  Find  the  value  of   x  in  these  equations : 

(1)  .35z  =  7;  (2).58z  =  11.6;  (3)  1.28*  =  3.84;  (4)  .092z  =  3.68; 
(5)  .019z  =  133;  (6)  7.51z  =  302.8;  (7)  .175z  =  10.5;  (8)  1.093z  = 
13.116. 

SOLUTION  of  (1):  x=-^-=20. 
.00 

10.  Writing  law  (1)  of  problem  8  in  the  form  .028Z  =  F,  find 
the  loads  (L)  to  2  decimals,  the  following  forces  (F)  will  draw 
under  the  conditions  of  problem  8  (1) : 

(1)  50  lb.;         (3)  56.28  lb.;         (5)     68.58  lb. ; 

(2)  135  lb.;         (4)  165£  lb. ;          (6)  110.35  lb. 

11.  To  find  the  loads  (L)  that  given  forces  (F)  will  draw,  the 
laws  of  problem  8  are  more  conveniently  writtenjthus :  L  =  ^-g  F= 
35.71  F.     Change  other  laws  of  problem  8  to  this  form. 


276  RATIONAL   GRAMMAR   SCHOOL   ARITHMETIC 

CONSTRUCTIVE  GEOMETRY 

§177.  Problems. 

PROBLEM  I. — Draw  a  perpendicular  to  a  given  line  from  a 
point  upon  the  given  line. 

EXPLANATION. — Let  AB  denote  the  given  line,  and  let  P  be  a  point 
upon  AB. 

We  wish  to  construct  a  perpendicular  to  AB  at  P.     Place  the  pin 
foot  on  P  and  close  the  compass  feet  until  their  distance  apart  is  less 

than  either  PA  or  PB.    Pi  is  such 

34.  a   distance.       Then    with    Pi    as 

/<D  radius   draw   a    short  arc  across 

AB  at  both  1  and  2. 

Now  spread  the  feet  a  little 
wider  than  Pi,  place  the  pin  foot 
on  1  and  draw  arc  3.  Arc  3  may 

) „  be  drawn  either  above  or  below  P. 

1\  P  I2~  ~~*  In  Fig.  140  it  is  above  P.     With- 

FIGUBE  140  OU^  change  of  radius  put  the  pin 

foot  on  2  and  draw  arc  4  across 
arc  3.     Let  D  be  the  intersection  of  arcs  #  and  4. 

With  the  straight  edge  of  your  ruler  draw  the  line  PD.     PD  is  the 
desired  perpendicular  to  AB. 

EXERCISES 

1.  Draw  straight  lines  on  your  paper,  mark  a  point  upon  each 
line  and  draw  perpendiculars  through  the  marked  points  until  you 
understand  fully  how  it  is  done. 

2.  Draw  the  perpendicular  to  any  line  you  may  draw  on  your 
paper  through  a  marked  point  on  the  line,  drawing  arcs  3  and  4 
below  the  line. 

3.  Draw  a  straight  line  on  the  blackboard,  mark  a  point  upon 
it,  and  with  crayon,  string,  and  ruler  draw  a  perpendicular  to 
the  line  through  the  marked  point. 

4.  Draw  a  straight  line  on    th    ground  and  along   the  edge 
of  a  board,  mark  a  point  on  it,  and  with  cord,  stake,  and  a  straight 
board,  draw  a  perpendicular  through  the  point. 

PROBLEM  II. — Draw  a  perpendicular  to  a  given  line  through 
a  point  outside  of  (not  on)  the  given  line. 


CONSTRUCTIVE    GEOMETRY 


277 


EXPLANATION.  —  Let  AB  denote  the  given  line  and  P  the  marked 
point  outside  of  AB.    We  wish  to  draw  a  perpendicular  toAB  through  P. 

Put  the  pin  foot  on  P  and  „ 

spread  the  feet  far  enough  apart 
so  that  the  pencil  foot  will  reach 
below  the  line  AB.  Then  draw  a 
short  arc  at  1  and  at  2. 

Put  the  pin  foot  on  1  and 
draw  arc  3.  Then  with  pin  foot 
on  2,  draw  arc  4  crossing  3  at  D. 

Connect  P  and  D  with  ruler 
and  pencil. 

PD  is  the  desired  perpendic-    -A  —  ::J^: 
ular. 

NOTE.—  The  arcs  S  and  4  might 
be  drawn  with  radii  of  any  length 
greater  than  half  the  distance 
from  1  to  2\  but  both  3  and  4 
must  be  drawn  with  the  same 
radius. 


•» 


FlGUEB  141 


EXERCISES 


1.  Draw  straight  lines  on  paper  or  on  the  blackboard,  mark 
points  outside  of  them,  and  draw  perpendiculars  to  the  lines,  until 
the  way  of  doing  it  is  fully  understood. 

2.  With  cord,  stake,  and  edge  of  a  board  draw  a  perpendicular 
on  the  ground  through  a  point  not  on  a  line.     (Fig.  47,  p.  107). 

III.  —  To    draw    a   square    inside   of    a    circle  of 


PROBLEM 
radius  \\"  . 


EXPLANATION.  —  Let  0,  Fig.  142,  be  the 
center  and  let  OB  =  {#'. 

Draw  a  diameter  AB. 

Spread  the  compass  feet  apart  wider 
than  ft"  and  putting  the  pin  foot  first  on  A, 
B  then  on  B,  draw  the  arcs  1  and  2.  Through 
E  and  O  draw  a  line  and  extend  it  both 
upward  and  downward  until  it  cuts  the 
circle  at  C  and  D. 

How  do  AB  and  CD  compare  in  length? 

With  ruler  and  pencil  connect  .Band  C; 
C  and  A  ;  A  and  D;  D  and  B. 

The  figure  EC  AD  is  the  desired  square. 


EXERCISES 


1.  Draw  a  square  inside  of  a  circle  of  1"  radius;   of 
diameter  \  of  If"  diameter. 


278 


RATIONAL    GRAMMAR    SCHOOL   ARITHMETIC 


FIGURE  143 
Old  Independence  Hall,  Philadelphia 


§178.  To  Model  a  3-Inch  Cube. 

The  cube  is  much  used  by  architects  in  buildings  and  parts  of 
buildings.     See  the  lower  part  of  the  tower  of  Old  Independence 

Hall,  Fig.  143.  Some  crystals  found 
in  nature  also  have  the  cubical 
form.  Can  you  suggest  any  uses 
or  occurrences  of  the  cube? 

Draw  the  development,  or  pat- 
tern, of  a  3"  cube. 

Draw  a  line  as  ac  nine  inches 
long  and  mark  it  off  into  3 -inch 
spaces.  Mark  the  points  of  divi- 
sion. At  p  draw  a  perpendicular 
line  to  #c,  prolong  the  perpen- 
dicular and  lay  off  3-inch  spaces 
on  it  as  in  Fig.  144.  Through 
b  draw  a  line  parallel  to  the  per- 
pendicular, dp,  as  in  Problem  II 
or  III,  p.  188,  or  Problem  VI, 
p.  191.  On  this  line  set  off  3-inch 
spaces  and  complete  the  develop- 
ment, as  shown  in  the  figure.  Leave  flaps  as  indicated. 
Crease  the  paper  along  the  lines  over 
the  edge  of  a  ruler  and  fold  up  a  3- 
inch  cube.  Paste  all  edges  excepting 
those  at  the  top. 

1.  What  is  the  area  of  each. sur- 
face? of  all  the  surfaces? 

2.  How  many  edges  has  the  cube? 
How  many  faces  meet  in  each  edge? 

The  surfaces  of  the  cube  meet 
each  other  in  the  edges  forming  lines. 

Call  the  corners  vertices  (ver'-ti- 
ses).  A  single  corner  is  a  vertex. 

3.  How  many  vertices  has  the  cube? 
How  many  edges  meet  at  each  vertex? 

The  edges  meet  each  other  in  the 
corners  of  the  cube  forming  points. 

4.  In  how  many  faces  does  each  vertex  lie? 

5.  Do  cubes  have  length?  breadth?   thickness?     Do  surfaces? 
I|o  lines?    Do  points? 


FIGURE  144 
Development  of  3"  cube 


CONSTRUCTIVE    GEOMETRY  279 

6.  What  are  the  limits  of  cubes?  of  surfaces?  of  lines? 

Patterns  of  the  surfaces  of  figures  like  those  of  Fig.  144  will 
hereafter  be  called  developments. 

7.  Place  an  inch  cube  in  the  corner  of  the  3-inch  cube.    How 
many  inch  cubes  will  fill  the  row  along  one  edge  of  the  larger 
cube? 

8.  How  many  such  rows  will  form  a  layer  on  the  bottom? 

9.  How  many  such  layers  will  fill  the  cube? 

10.  How  many  inch  cubes  will  fill  a  3-inch  cube? 

11.  How  many  cubic  inches  are  there  in  a  4-inch  cube?  in  a 
10-inch  cube?  in  an  or-inch  cube? 

§179.  To  Model  a  Square  Prism. 


m 


Draw  a  straight  line  ad  (Fig. 
1 45)  12  inches  long,  and  make  ab  and 
dc  each  2  inches  long.  Draw  en  and 
fo  perpendicular  to  ad  at  b  and  c. 
Lay  off  distances  eb,  bg,  gl,  ln,fcy  cli, 
lim,  mo,  each  equal  to  2  inches. 
Draw  ef,jk,  etc.,  and  complete  the 
development.  Provide  flaps;  cut 
and  paste  the  model.  FIGURE  ir, 

Development  of  the  Square  Prism 

1.  What  is  the  area  of  each  of  the  end  surfaces? 

2.  Give  the  area  of  each  side  surface;  of  all  the  side  surfaces. 

3.  How  many  inch  cubes  will  the  model  hold? 

4.  By  measuring  the  edges  how  could  you  find  the  number  of 
cubic  inches  the  model  will  hold? 

5.  How  many  cubic  inches  would  it  hold  if  it  were  1  inch 
longer?  twice  as  long? 

6.  If  every  line  in  the  pattern  were  twice  as  long  as  in  the 
development,  how  would  you  answer  questions  1  to  4? 

Pupils  should  make  their  own  models,  different  pupils  using 
different  lengths  of  lines  and  even  devising  different  forms  of 
pattern.  For  example,  let  one  pupil  use  a  3-inch  fundamental 
distance  instead  of  an  inch,  another  a  4-inch  distance,  and  so  on, 
each  comparing  his  completed  model  with  other  models. 


280 


RATIONAL   GRAMMAR   SCHOOL   ARITHMETIC 


§180.  To  Model  a  Flat  Prism. 


FIGURE  146 
Developmeut.of  Flat  Prism 


Draw  a  line  ae  (Fig.  146)  18  in. 
long  and  mark  off  ed=3in.,  dc  = 
6  in.,  cb  =  3  in.,  and  la  =  6  inches. 

At  c  and  d  draw  perpendiculars 
to  ae  and  extend  them  11  in.  below 
c  and  d  to  r  and  s,  and  3"  above  6' 
and  d  to  &  and  /. 

Make  co  and  dp  8  in.  long  and 
draw  mq  through  o  and  p. 

Complete  the  development  by 
drawing  the  necessary  parallels. 

Provide  it  with  the  necessary 
flaps  and  paste  up  the  model,  leaving 
the  top  ckdl  open. 


1.  With  the  aid  of    the    model   of    an  inch  cube,  find  how 
many  cubic  inches  would  fill  the  flat  prism. 

2.  To  what  is  the  product  of  the  3  different  edges  equal? 

3.  How    could    you  obtain   the   capacity  in  cubic  inches  of 
such  a  }>risin  by  measuring  the  lengths  of  its  edges? 

4.  How  many  square  inches  in  the  whole  surface  of  the  flat 
prism  of  Fig.  14G? 


§181.  Comparison  of  Prisms. 

(a)  A  right  prism. 

Using  the  lengths  of  lines  as 
shown  in  Fig.  147,  draw  a  pattern  on 
heavy  paper  and  construct  the  model 
of  a  right  prism  as  was  done  in 
Fig.  145.  Provide  the  edges  with 
flaps  and  paste  up  the  model,  leav- 
ing one  end  open,  so  that  it  may  be 
filled  with  sand.  How  many  cubic 
inches  of  sand  will  just  fill  the  model? 

How  can  you  find  the  capacity 
of  the  model  by  using  the  lengths 
of  the  edges? 


FIGURE  147 
Development  of  Right  Prism 


CONSTRUCTIVE    GEOMETRY 


281 


(b)  An  oblique  prism  with  parallelograms  for  bases. 

Similarly  draw  a  pattern,  cut,  

fold,  and  paste  up  a  model  of  the 
oblique  prism,  as  shown  in  Fig. 
148,  leaving  an  end  unpasted  for 
sand. 

When  the  model  of  the  right 
prism  is  just  full  of  sand,  the 
sides  not  being  bulged  out  with 
the  sand,  pour  the  sand  into  the 
model  of  the  oblique  prism.  Does 
it  fill  the  second  model? 

What  is  the  ratio  of  the  capac- 
ities of  the  two  models? 

What  is  the  ratio  of  the  areas 
of  the  bases  (ends)? 

(c)  A    triangular  prism. 


FIGURE  148 
Development  of  Parallelogram  Prism 


X 


Model  a  triangular 
prism  like  the  one  shown 
in  Fig.  149. 

Compare  the  capacity 
of  this  model  with  that 
of  the  model  of  a  square 
prism  I"xl"x4".  What 
is  their  ratio? 

Model     a     triangular 
prism    such   as    that    of 
Fig.    149,    but    use    for 
lengths  3"  instead  of  1" 
and    7"   instead   of    4". 
How  does  its  capacity  compare  with  that  of  the  prism  of  Fig.  147? 
of  Fig.  148?  Give  the  ratio  in  each  case. 
Compare  the  ratios  of  the  bases. 

§182.  Volume  of  an  Oblique  Prism. 

The  volume  of  a  figure  is  the  number  of 
cubical  units  in  the  space  enclosed  by  its 
bounding  surfaces. 

1.  How  many  cubic  inches  are  there 
in  a  straight  pile  of  visiting  cards  2"  high,  if 
each  card  is  2"  x  3"  ? 


FIGURE  149 
Development  of  Triangular  Prism 


FIGURE  iso 

P  Right 


HAT1UJNAL    liKAMMAR    SCHOOL    ARITHMETIC 


2.  How  many  cubic  inches  would  there  be  in  the  pile  if  it 
were  5"  high?  9"  high?  a  in.  high? 

3.  Push  the  straight  pile  of  problem  1  over  as  in  Fig.  151. 
How  many  cubic  inches  of  paper  are  there  in  this  oblique  pile? 

.  4.  Has  the  height  of  the  pile  been 
changed  in  Fig.  151?  has  the  area  of 
the  base?  has  the  volume? 

5.  How  can  you  find  the  number  of 
cubic  units  in  a  right  prism  from  the 
FIGURE  i5i  area  of  its  base  and  its  height?  in  an 

Oblique  Pile,  or  Oblique  Prism        oblique  prism? 

6.  Find  the  volumes  of  square  prisms  having  edges  of  the 
following  lengths  : 

(1)  3"  x  5"  x  6"  ;     (4)  4f  '  x  6'       x  12'  ;     (7)  a  in.  x  b  in.  x  c  in.  ; 

(2)  6"  x  5"  x  9"  ;     (5)   16'  x  lOf  '  x  20'  ;     (8)  x  ft.  x  v  ft.  x  z  ft.  ; 
(8)  15"x9"x8";     (6)  45"  x  16f"  x  21";    (9)  -myd.  x  nyd.  xp  yd. 


§183.  Paper-Folding, 

For  the  following  exercises  in  paper-folding  any  moderately 
thick,  glazed  paper  will  do.  Tinted  or  colored  paper,  without 
lines,  will  however  show  the  creases  more  clearly.  It  is  con- 
venient to  have  the  paper  cut  into  pieces,  about  4"  square.  Such 
paper  is  inexpensive  and  may  be  had  of  any  stationery  dealer. 

PROBLEM  I. — At  a  chosen  point  on  a  line,  make  a  perpendicular 
to  the  line,  by  folding  paper. 

EXPLANATION. — Fold  one  part  of  a  piece  of  paper  over  upon  the  other 
and  crease  the  paper  along  the  fold,  as  at  AB,  by  drawing  the  finger  along 
the  fold.  Taking  D  to  denote  the  chosen  point,  fold  the  paper  over  the 
point  D,  and  bring  the  two  parts,  DA  and  DB,  of  the  crease  AB 
exactly  together.  Hold  the  paper  firmly  in  this 
position  and  crease  the  paper  along  the  line  DC. 

Compare  the  portion  of  the  paper  between  the 
creases  DB  and  DC  with  the  portion  between  the 
creases  DA  and  DC.  How  do  the  angles  BDC  and 
CDA  compare  in  size? 

DEFINITIONS. — When  two  lines  meet  in  this  way 
making  the  angles  at  their  point  of  meeting  (inter- 
section) equal,  the  lines  are  said  to  be  perpendicular 
to  each  other,  and  each  is  called  a  perpendicular  to 
the  other. 

The  angles  thus  formed  are  called  right  angles.  FIGURE  153 


CONSTRUCTIVE    GEOMETRY 


283 


PROBLEM  II. — Bisect  an  angle,  by  folding  paper. 


FIGURE  153 


EXPLANATION. — Crease  two  lines,  as  OA  and  OB, 
lying  across  each  other.  They  make  the  angle 
AOB. 

Now  fold  the  paper  over,  and  bring  the  crease 
OA  down  on  OB.  Holding  the  creases  firmly  to- 
gether crease  the  bisector  OC. 

How  do  the  angles  AOC  and  BOC  compare  in 
size?  Do  they  fit? 

What  is  the  ratio  of  angle  AOC  to  angle  BOC? 
of  AOB  to  AOC? 

How  could  the  angle  AOB  be  divided  by  creases 
into  4  equal  parts? 


'.  !,. :.':!. -''."Tig 


PROBLEM  III. — Crease  three  non-parallel  lines. 

EXPLANATION. — Crease  AB  in  any  position.  Crease  CD  in  any  posi- 
tion, not  parallel  to  AB.  Finally,  crease  EF  in  any  position  not  parallel 
to  either  AB  or  CD,  Fig.  154. 

In  general,  in  how  many 
points  do  three  non-parallel 
lines  cross  each  other? 

Can  you  crease  three  lines, 
no  two  of  which  are  parallel, 
in  such  a  way  as  to  obtain  just 
two  crossing  points  (intersec- 
tions)? Try  it. 

Crease  three  non-parallel 
lines  in  such  positions  as  to 
give  but  one  intersection  (see 
Fig.  155).  Lines  which  go 
through  the  same  point  are 
called  concurrent  lines. 


FIGURE  154 


FIGURE  155 


PROBLEM    IV. — Crease    the   bisectors    of   the  3  angles  of   a 
triangle. 


EXPLANATION. — Crease  out  a  triangle  such  as 
ABC. 

Then  crease  the  bisector  of  each  angle,  as  in 
Problem  II.  Work  carefully. 

1.  How  do  the  bisectors  cross  each  other? 

2.  Since  the  three  bisectors  of  the  three 
angles  of  a  triangle  all  go  through   0,  what 
name  would  be  applied  to  them 


FIGURE  156 


284  RATIONAL   GRAMMAR    SCHOOL   ARITHMETIC 

PROBLEM  V. — Bisect  a  given  line,  by  paper-folding. 

EXPLANATION. — Crease  <*  line  and  stick  the  point 
of  a  pin  through  the  crease  at  A  and  at  B. 

AB  is  the  line  to  be  bisected. 

Fold  the  paper  over  and  bring  the  pin  hole  at 
B  down  on  the  pin  hole  at  A.  Press  the  paper 
down  and  crease  a  line  across  AB,  as  at  C.  CD  is 
the  perpendicular  bisector  of  AB. 

Crease  a  line  from  D  to  B  and  another  from  D  to 
A.  When  the  paper  is  folded  over  the  line  CD,  how 
do  these  two  creases  seem  to  lie?  Compare  the 
lengths  of  DB  and  of  DA. 

1.  Crease  the  perpendicular  bisectors  of  the  3  sides  of  a  triangle 
and  find  how  they  cross  each  other. 

2.  What  kind  of  lines  are  the  perpendicular  bisectors  of  the 
sides  of  a  triangle? 

PROBLEM  VI. — Crease  a  square  and  its  diagonals. 


EXPLANATION.— Crease  2  lines,  as  AB  and  AX, 
perpendicular  to  each  other.  (See  Problem  I.) 

Fold  the  paper  over  the  point  B  and  when  the 
two  parts  of  the  crease  AB  fit,  crease  the  line  BY. 

Now  fold  the  paper  over  so  that  crease  BY 
conies  down  along  crease  BA,  and  crease  the  line 
BC. 

Fold  the  paper  over  a  line  through  C,  bringing 
CX  down  along  CA,  and  crease  CD. 

ABCD  is  the  required  square. 

FIGURE  158 

1.  When  the  square  is  folded  over  the  diagonal  BC,  where 
does  D  fall?     Along  what  crease  does  BD  lie?  CD? 

2.  How  does  the  diagonal  divide  the  area  of  the  square? 

3.  Fold  over  and  crease  the  diagonal  AD?     How  does  it  divide 
the  square? 

4.  Compare  OB  with  OC\  OD  with  OA. 

5.  How  do  the  diagonals  of  a  square  divide  each  other? 

6.  How  do  the  diagonals  divide  the  area  of  the  square? 

7.  How  does  AD  divide  the  angle  BA  C?  the  angle  JSDC? 

8.  Fold  a  second  square  and  crease  its  diagonals  (Fig.  159). 
Fold  over  0,  bringing  D  down  on  HO  and  crease  HG.    Similarly 
crease  EF, 


CONSTRUCTIVE    GEOMETRY 


285 


9.  When  the  paper  is  folded  over  OF  along  what  line  does 
Onfall?  Offl 

10.  How  does    OB  compare  with  OD  in 
length?  How  then  do  AD  md  B C  compare? 

11.  How    do    EF  and  HG    divide    the  0 
square? 

12.  Crease  the  following  lines :   GfF,  FH, 
HE,  and  EG.    How  does  the  area  of    the 

square  GFHE  compare  with  that  of  ABDC?  FIGURK  ir/j 

PROBLEM   VII. — Crease   the   three   perpendiculars  from  the 
vertices  of  a  triangle  to  the  opposite  sides. 

EXPLANATION.— Crease  the  triangle  ABC,  Fig. 
160.  Fold  the  paper  over  the  vertex  C,  and  bring 
the  crease  DB  down  along  DA,  and  crease  the  per- 
pendicular CD. 

Similarly  crease  AE  and  BF.  Work  carefully. 
How  do  the  creases  CD,  AE  and  BF  cross  each 
other? 

What  kind  of  lines  are  the  three  perpen- 
diculars from  the  vertices  to  the  opposite  sides  of 
a  triangle? 


FIGURE  160 

PROBLEM  VIII. — Crease  the  three  perpen- 
dicular bisectors  of  the  sides  of  a  triangle, 
Fig.  161. 

How  do  they  cross?  What  kind  of  lines 
are  they? 

PROBLEM  IX. — Crease  a  rectangle  and  its 
diagonals. 


FIGURE  161 


EXPLANATION.— Crease  AB,  Fig.  162,  making  the 
distance  from  A  to  B  two  inches.  By  Problem  I, 
crease  the  perpendiculars  AD  and  BC.  Make  AD 
and  BC  each  1"  long  and  crease  CD. 

Crease  the  diagonals,  one  through  A  and  C,  and 
the  other  through  B  and  D.  Call  their  crossing 
point  O. 

Fold  the  paper  over  the  perpendicular  FE,  bring- 
ing B  down  on  A.  Where  does  C  fall?  What 
other  line  equals  2J5?  OB?  OC? 

Fold  the     aper  over  the  point  O  so  that  B  falls 
on   C,  and  cfeas-   the  perpendicular  HG.      What 
FIGURE  162  other  line  equals  AB1  OG1  OD1 

How  does  the  intersection,  O,  of  the  diagonals  divide  the  diagonals? 


/COD  KAT1UJNAL,    WKAMMAK    SCHOOL    ABITHMETIC 

§184.  Perimeters.— The  perimeter  of  any  figure  is  the  sum  of  the 
lines  bounding  the  figure.  Thus,  in  form  (6),  Fig.  163,  if  p  de- 
note the  perimeter, 

(i)  p=x+y+  z- 

1.  What  does  x  +  2x  +  ±x  mean?     How  may  it  be  more  briefly 
written? 

2.  What  is  the  coefficient  of  x  in  the  answer  to  problem  1? 

3.  Write  the  perimeter,  p,  of  (1),  Fig.  1G3,  in  two  ways. 

4.  Write  the  perimeter,  p,  of  (13)  in  three  ways. 

5.  Write  the  perimeter,  p,  of  (12)  in  three  ways. 

6.  Write  -J  of  the  perimeter,  p,  of  (13)  in  two  ways. 

7.  Write  an  expression  showing  that  the  two  answers  to  6  are 
equal. 


FIGURE  163 

8.  In  (12),  if  x  =  80  rods  and  y  =  40  rods,  how  many  feet  in 
the  perimeter? 

9.  In  (1)  and  (2),  if  x  and  y  each  =  20  rods  and  one  side  of 
the  triangle  rests  on  the  square,  making  a  new  figure,  omitting 
the  common  line,  what  is  the  perimeter  of  this  new  form,  in  feet? 
How  many  sides  has  the  new  figure  thus  formed? 

10.  Forms   (1)    and  (13)   are  combined  into  a  single  figure. 
Write  p  for  the  new  figure  in  two  ways,  supposing  x  the  same  in 
both. 


CONSTRUCTIVE    GEOMETRY  287 

§185.  auadrilaterals. 

1.  Forms  (2),  (7),  (8),  (10),  (11),  (12),  and  (13),  Fig.  163,  are 
different  kinds  of  quadrilaterals.     What  is  a  quadrilateral  ? 

2.  What   quadrilaterals   have    their    opposite   sides    parallel? 
These  figures  are  parallelograms.     Define  a  parallelogram. 

3.  What  parallelograms  have  all  their  sides  equal?    What  is  a 
rhombusf 

4.  What  rhombus  has  all  its  angles  equal?     What  is  a  square? 

5.  What  quadrilaterals  have  their  opposite  sides  equal  but  con- 
secutive angles  not  equal?     Define  a  rhomboid. 

6.  What  parallelograms  have  their  angles  all  equal?    Define  a 
rectangle. 

7.  What  quadrilateral  has  only  one  pair  of  sides  parallel?     De- 
fine a  trapezoicl. 

8.  Is  a  trapezoid  a  parallelogram?     Is  it  a  quadrilateral? 

9.  Is  a  rectangle  necessarily  a  quadrilateral?   Is  it  a  parallel- 
ogram? a  square?     May  a  rectangle  be  a  square? 

10.  Is  a  square  necessarily  a  quadrilateral?    Is  it  a  parallel- 
ogram? a  rectangle?  a  rhombus? 

§186.  Perimeters  of  Miscellaneous  Figures. 

1.  Denote  the  perimeter  of  each  of  the  forms  in  Fig.  163  by 
p  and  write  an  equation  like  (I)  in  §184,  showing  the  value  of  p 
for  each  figure. 

2.  Omitting  the  lines  on  which  the  forms  join,  write  an  equa- 
tion  showing   the  value   of  p  when    (2)  and  (6),   Fig.   163,  are 
joined  on  y,  which  has  the  same  value  in  both. 

3.  In  the  same  way  join  (3)  and  (10)  on  «,  which  has  the  same 
value  in  each,  and  write  an  equation  showing  the  value  of  p. 
What  is  the  name  of  the  figure  thus  formed? 

4.  Join  (2)  and  (7)  in  which  x  and  y  are  equal.    If  the  perimeter 
of  the  figure  thus  formed  is  240  rods,  find  the  value  of  x  in  feet. 

5.  What  is  the  length  of  the  perimeter  of  a  figure  like  (6),  Fig. 
163,  whose  sides  are  2  in.,  a  in.,  and  1)  in.  long?  whose  sides  are 
x  in.,  8  in.,  and  z  in.  long?  like  (10),  whose  sides  are  c  ft.,  2c  ft., 
d  ft.,  and  x  ft.?  like  (12),  whose  sides  arc  2z  rd.,  4y  rd.,  2#  rd., 
and  4=y  rd.  long? 


Write  the  perimeter  (j;)  of  each  of  the  figures  below. 


y 

y 

(0) 

—  o  — 

-    --  ^ 

i 

X 

X 

1 

(7) 

y 

y         o 

_j  ^ 

X 


(9} 


FIGURE  164 


' 

d            d 

n 

(10) 

•i 

6  

In  such  forms  as  those  from  '5)  to  (10),  Fig.  164,  some  line, 
or  lines,  must  be  found  by  subtracting  others.  In  (G)  for  example, 
note  that  the  ends  are  each  x  —  y.  The  perimeter  is  then  x  +  x-\- 
(x-y)+y  +y  +  (x-y)  =  ±x -y  -y  +  y  +  y  =  ±x- %  +  2y=  4rc. 

All  sides  not  lettered  must  be  expressed  without  using  other 
letters  than  those  given  on  the  figure.  The  perimeter  means  the 
sum  of  all  the  lines  that  bound  the  strip,  or  surface,  of  the 
figure. 

NOTE. — One-half  the  difference  of  a  line  ra  and  a  line  n  is  written 

m  —  n 

— ^ — or  \  (m  —  n). 

1.  In  (8),  a  =  15',  I  =  12',  x  =  25'  and  v  =  30'.    Find  the  length 
of  the  perimeter  of  the  figure. 

2.  Find  the  area  of  (8)  enclosed  by  the  solid  lines. 

3.  Make  and  solve  other  similar  problems. 


CONSTRUCTIVE    GEOMETRY 


289 


FIGURE  165 


:187.  Measuring  Angles  and  Arcs. 

ORAL   WORK 

We  may  measure  the  amount  of  turning  of  each  clock  hand  in 
either  of  two  ways : 

(1)  By  the  length  of  the  circular  arc 
passed   over  by   the   tip   of    the   rotating 
hand ; 

(2)  By  the  wedge-shaped  space  having 
its    point    at    the    hand-post,     A,     over 
which  the  stem,   A-III,    A-VI,    etc.,   of 
the  rotating  hand  moves. 

While  the  clock  hand  turns  from  XII 
to  III  its  tip  moves  over  1  quadrant  of  arc 
and  its  stem  moves  over  a  right  angle. 

1.  While  the    hand  turns  from  XII  to  VI  over  how    many 
quadrants  does  its  tip  move?  Over  how  many  right  angles  does  its 
stem  turn? 

2.  Answer  same  questions  for  a  turn  of  the  hand  from  XII  to 
IX ;  from  XII  to  XII  again ;  from  XII  around  through  XII  to  III. 

3.  Over  what  part  of  a  right  angle  does  the  stem  of  the  hand 
move  while  the  hand  is  passing  from  XII  to  I?  from  XII  to  II? 
from  III  to  IV?  from  VI  to  VIII? 

4.  Over  what  part  of  a  quadrant  does  the  tip  of  the  htind  pass 
in  each  case  of  problem  3? 

5.  Over  what  part  of  a  right  angle  does  the  stem  of  the  hand 
move  while  the  tip  moves  1  min.  along  the  arc?     In  the  same 
case  over  what  part  of  a  quadrant  does  the  tip  move? 

DEFINITIONS. — Any  wedge-shaped  part  of  the  face  m<  yed  over  by 
the  stem  of  a  clock  hand  as  it  turns  around  the  hand-post  is  called  an 
angle.  The  curve  passed  over  by  the  tip  of  the  hand,  while  the  stem  of 
the  hand  moves  over  the  angle,  is  called  the  arc  of  the  angle. 

ILLUSTRATIONS. — The  wedge-shaped  spaces,  having  their  points  at  A, 
and  included  between  any  two  positions  of  the  hand,  as  A-XII  and  A-I, 
A-I  and  A-III,  A-I  and  A-VI,  are  all  angles ;  the  space  swept  over  by  the 
hand,  A-I,  as  it  moves  around  through  II,  III,  IV,  V,  VI,  etc.,  to  IX  is 
also  an  angle. 

6.  As  the  hand  moves  from  A-XII  to  A-VI,  that  is,  so  that  the 
two  positions  of  the  hand  are  in  the  same  straight  line,  the  hand 
moves  over  a  straight  angle.     A  straight  angle  equals  how  many 
right  angles? 


290  RATIONAL   GRAMMAR   SCHOOL   ARITHMETIC 

7.  The  arc  of  a  straight  angle  equals  "how  many  quadrants? 
How  many  quadrants  make  a  complete  circle? 

8.  How   many  5-min.   spaces   make   the   circumference   of   a 
complete  circle?     How  many  1-min.  spaces  make  the  circumfer- 
ence of  a  complete  circle? 

9.  What  part  of  a  right  angle  is  passed  over  by  the  stem  of  the 
hand  as  its  tip  moves  from  one  end  to  the  other  of  a  1-min.  space? 

10.  If  lines  were  drawn  from  the  hand-post  to  the  ends  of  all 
the  1-min.  spaces,  the  whole  face  of  the  clock  would  be  divided 

up  into  how  many  small 
angles? 

The  instrument  in 
common  use  for  meas- 
uring angles  and  arcs 
is  the  protractor.  (See 
Fig.  166.) 

11.  Instead  of  divid- 
ing the  right  angle  up 
FIGURE  t66  by  radiating  lines  into 

15  equal  parts,  the  protractor  divides  the  right  angle  into  90  equal 
parts,  one  of  which  is  called  the  angular  degree.  These  same  lines 
would  divide  the  quadrant  up  into  how  many  equal  parts?  Each 
of  these  parts  is  a  degree  of  arc. 

12.  How  many  angular  degrees  are  there  in  a  straight  angle? 
How  many  degrees  of  arc  in  the  arc  of  a  straight  angle? 

13.  How  many  angular  degrees  are  swept  over  by  a  clock  hand 
while  moving  entirely  around  once?     How  many  degrees  of  arc  in 
the  circumference  of  a  circle? 

14.  How  many  degrees  of  angle  are  passed  over  by  the  stem  of 
the  minute  hand  in  two  hours?     In  the  same  time  how  many 
degrees  of  arc  are  passed  over  by  the  tip  of  the  hand? 

15.  A  sextant  is  J  of  a  circumference ;  how  many  degrees  of 
arc  are  there  in  a  sextant? 

16.  An  octant   is   -J  of   a  circumference;  how  many  degrees 
of  arc  are  there  in  an  octant? 

17.  Study  the  protractor,  Figs.  166  and  167;  notice  how  its 
marks  are  numbered.    How  many  degrees  of  arc  are  there  between 


CONSTRUCTIVE    GEOMETRY 


291 


the  closest  lines  on  the  outer  edge?    How  many  degrees  of  angle 
are  there  between  the  lines  which  converge  toward  the  center? 

18.  For  smaller  angles  a  shorter  unit  is  -fa  of  a  degree  of  angle 
and  the  smaller  unit  is  called  the  minute  of  angle.    ¥V  of  the 


FIGURE  167 

minute,  called  the  second,  is  a  still  smaller  unit.  How  many 
minutes  of  angle  in  a  right  angle?  in  a  straight  angle?  in  a  com- 
plete revolution,  or  perigon?  How  many  seconds,  in  each  case? 

19.  The  arcs  between  the  sides  of  the  minute  and  the  second 
angles  are  the  minute  and  the  second  of  arc.  How  many  minutes 
of  arc  in  a  quadrant,  in  a  sextant,  in  an  octant,  in  a  circumfer- 
ence? How  many  seconds,  in  each  case? 

A  paper  protractor  can  be  purchased  at  any  bookstore,  and 
each  pupil  should  supply  himself  with  one. 

To  measure  an  angle  the  center  of  the  protractor  is  placed  on 
the  vertex  A,  and  the  0  line  through  A  is  placed  along  one  side 
AD  (see  Fig.  168) .  The  reading  of  the  mark  below  which  the  other 


FIGURE  168 


side,  AF,  falls  is  the  number  of  degrees  in  the  angle,  or  in  the 
arc  of  the  protractor  included  between  AD  and  AF. 


292  RATIONAL   GRAMMAR   SCHOOL   ARITHMETIC 

WRITTEN    WORK 

20.  Draw  a  triangle,  and,  with  a  protractor,  measure  its  angles. 
To  what  is  the  sum  of  all  three  equal? 

21.  With  ruler  draw  a  half  a  dozen  triangles  of  different  shapes, 
carefully  measure  each  angle,  and   find   the  sum  of   the   three 
for  each  triangle. 

22.  Measure  with  the  protractor  the  number  of  degrees  in  the 
angle  at  one  corner  of  the  page  of  this  book. 

23.  Draw  two  straight  lines  crossing  each  other  as  in  Fig.  169. 

With  the  protractor  measure  and  compare  a  and  c  ;  b 
and  d. 


/  The  angles  a  and  c,  or  5  and  d,  lying  opposite 

each   other,    are   called    opposite  angles,   or  vertical 
FIGURE  169     angles. 

24.  Draw  two  crossing  lines  in  different  positions,  and  in  each 
case  compare  the  measures  of  a  pair  of  opposite  angles.     What 
do  you  find  to  be  true? 

25.  Draw   a   pair   of    parallel   lines,  as   a   and    Z»,  Fig.    170, 
and  draw  a  third  straight  line  c  cutting   across  the 
parallels.     Measure  with  a  protractor  and  compare  the 

four  angles,  in  which  1  is  written  in  Fig.  170.    Measure 
and  compare  those  in  which  2  is  written.     What  do 

FIGURE  170 

you  find  to  be  true? 

26.  Draw  a  line  perpendicular  to  another,  see  §177,  and  measure 
all  the  angles  formed.     What  do  you  find? 

DEFINITIONS.  —  The  lines  which  include  the  angle  are  sides  of  the  angle. 
The  point  where  the  sides  meet  is  the  vertex  of  the  angle. 

27.  Draw  a  quadrilateral  of  any  irregular  shape  and  measure 
each  of  its  4  angles.     To  what  is  the  sum  equal?     Try  another 
quadrilateral  and  see  whether  you  obtain  the  same  sum. 

28.  Draw  a  given  angle  at  a  point  on  a  gi^en  line. 

EXPLANATION.  —  Let  the  given  line  be  ED,  the  given  angle  35°,  and  the 
given  point  A,  Fig.  168.  Place  the  protractor  with  its  center  at  A,  and 
with  the  diameter  of  the  protractor  along  the  line  ED.  Make  a  dot 
opposite  the  35°  mark  on  the  protractor.  With  the  ruler  draw  a  straight 
line  through  A  and  this  dot  to  F.  The  angle  DAF  is  the  required  angle. 

29.  Draw  a  straight  line,  mark  a  point  on  the  line  and  draw 
the  angle  75°;  95°;  100°;  135°;  55°30'. 


CONSTRUCTIVE    GEOMETRY 


293 


TABLE  OF  UNITS  OF  ANGLE  AND  ARC  MEASUREMENT 
60  seconds  (")  =  1  minute  (') 
60  minutes        =  1  degree  (°) 
360  degrees         =  1  circumference  (or  perigon) 
90  degrees         =  1  quadrant  (or  right  angle) 


TABLE  OF  EQUIVALENTS 
1,296,000"  ] 

21,600' 
360° 
4  quadrants 


=  1  circle 


§188.   The  Sum  and  Difference  of  Angles. 

EXERCISE  1. — With  ruler  and  compasses,  draw  AB  and  CD 
perpendicular  to  each  other. 

1.  Suppose    a   protractor   supplied 
with  a  pointer  as  shown  in  Fig.  171, 
the   center   of    the    protractor    heing 
placed  at  the  point,  0,  over  how  many 
degrees   of  arc  would  the  tip   of    the 
pointer  turn   while   the   stem  of    the 
hand  turns  from  line    OB  to  line  0(7? 
from  OB  through  00  to  0^4? 

2.  How  many  degrees  are  there  in 
the  angle  BOC?    in  angle   #0^4?    in 
GOD1  in  A  OD?  in  the  angle  from  OA 
around  through  OD  to  OB?  from  OA 
around    to    00?     from     OA    entirely 
around  to  OA  again? 

3.  How  many  degrees  of  angle  fill   the  space  around  a  point, 

as  0,   on  one  side  of  a  straight  line, 
as  AS?  on  both  sides? 


DEFINITIONS. — An  angle  that  is  smaller 
than  a  right  angle  is  called  an  acute  angle 
(see  Fig.  172).  An  angle  that  is  larger  than 
a  right  angle  is  called  an  obtuse  angle. 


Acute  Angles        Obtuse  Angles 
FIGURE  172 


294 


RATIONAL    GRAMMAR    SCHOOL    ARITHMETIC 


FIGURE  173 


EXERCISE  2. — Find  the  sum  of  two  angles 

1.  Draw  two  acute  angles  like  those  at  the  top  of  Fig.  173,  care- 

fully cut  them  out,  and  place  them  as 
in  the  lower  part  of  the  figure.  Push 
their  vertices  and  nearer  sides  en- 
tirely together.  Draw  two  lines  along 
the  other  sides  of  the  angle.  Eemove 
the  angles  and  have  an  angle  like  that 
on  the  left,  which  is  the  sum  of  the 
two  [angles. 

2.  Calling  the  larger  angle  ic,  and  the  smaller  y,  what  denotes 
the  last  angle? 

3.  Draw  an  angle  equal  to  the  sum  of  an  acute  and  an  ohtuse 
angle. 

EXERCISE  3. — Find  the  difference  of  two  angles. 

1.  Draw  two  angles  like  those  at 
the  top  of  Fig.  174,  cut  them  out,  and 
place  them  as  in  the  lower,  left-hand 
part  of  the  figure.  Mark  along  the 
lower  side  of  the  upper  angle  and 
cut  along  the  mark.  The  lower  part 
of  the  larger  angle  (shown  on  the 
right)  equals  the  difference  of  the  two 
angles. 


FIGURE  174 


FIGURE  175 


FIGURE  176 


2.   If  the  larger  angle  equals  x  degrees  and  the  smaller  equals 
y  degrees,  how  many  degrees  are  there  in  the  diiference? 


CONSTRUCTIVE    GEOMETRY 


295 


FIGURE  177 


DEFINITIONS. — Two  angles  whose  sum  equals 
a  right  angle,  or  90°,  are  called  complemental 
angles  (Fig.  175).  Two  angles  whose  sum 
equals  2  right  angles,  or  180°,  are  called  sup- 
plemental angles  (Fig.  176). 

The  sum  of  all  the  angles  that  just  cover 
the  plane  on  one  side  of  a  straight  line  is  equal  «— . 
to  how  many  right  angles  (Fig.  177)?    To  how 
many  degrees  is  this  sum  equal? 

EXERCISE  4.— Draw  an  equilateral  triangle  (Problem  VII,  p. 
108).  Tear  off  a  corner  and  fit  it  over  each  of  the  other  corners  in 
turn.  How  do  the  three  angles  compare  in  size.  Tear  off  the 
other  corners  and  place  them  as  in  Fig.  178.  To  what  is  the  sum 
of  all  three  angles  equal? 

If  one  of  the  angles  is  x  degrees,  how  many  degrees  are  there 
in  the  sum  of  all  three  angles?  As  the  sum  of  the  three  angles 


FIGURE  178  FIGURE  179 

equals  both  3x  degrees  and  also  180°,  we  may  write  the  equation 

3x  =  180. 

If  3x  =  180,  to  what  is  x  equal?  How  many  degrees  are  there 
in  one  of  the  angles  of  an  equilateral  triangle? 

EXERCISE  5. — Draw  any  scalene  triangle.  Tear  off  the  corners 
and  place  them  as  in  Fig.  179.  To  how  many  right  angles  is  the 
sum  of  all  the  angles  of  the  triangle  equal?  To  how  many 
degrees? 

1.  Letting  #,  y,  and  z  denote  the  numbers  of  degrees  in  the 
respective  angles,  in  what  other  way  may  we  write  the  sum  of 
all  three?     What  equation  may  we  then  write?      Tell  what  the 
equation  #  +  ?/  +  £  =  1800  means. 

2.  Draw  other  scalene  triangles,  tear  off  the  corners,  place  them 
as  in  the  figure,  and  find  whether  x  +  y  +  z  =  180°  in  all  cases. 


3.  Crease  a   right  triangle   and   find   whether   the  equation, 
x  +  y  +  z=  180°,  is  true  for  it. 

4.  Cut  along  the  creases  and  tear  off  the  two  acute  angles  of  a 
carefully  creased  right  triangle  and  fit  them  over  a  carefully  creased 
right  angle?  What  seems  to  be  true?  If  this  is  true  what  equation 
may  you  write  for  the  sum  of  the  two  acute  angles  (x  and  y)  of  a 
right  triangle? 

EXERCISE  G. — Draw  a  quadrilateral  and  cut  it  along  a  straight 
line  from  one  corner  to  the  opposite  corner  as  in  Fig.  180.  Such 
a  line  as  is  indicated  by  the  cut  is  called  a  diagonal. 

1.  The  cut  divides  the  quadrilateral 
into  figures  of  what  shape? 

2.  To  what  is  the  sum  of  the  three 
angles  of  each  part  equal? 

3.  To  what  is  the  sum  of  all  six 


FIGURE  180 
angles  of  the  two  triangles  equal? 

4.  To  what  is  the  sum  of  all  4  angles  of  the  quadrilateral  equal? 

5.  Do  your  answers  hold  true  for  the  parallelogram  of  Fig.  180? 

6.  Do  they  hold  good  for  any  shape  of  quadrilateral  you  can  draw? 

7.  To  what  then  is  the  sum  of  the  four  angles,  #,  y,  z,  and  w, 
of  any  quadrilateral  equal? 

EXERCISE  7. — Draw  a  hexagon  and  cut  it  along  diagonals  as 
shown  in  Fig.  181. 

1.  Into  figures  of  what  shape 
is  the  hexagon  divided? 

2.  Into  how  many  such  figures 
do  the  diagonal  cuts  divide  the 
hexagon? 

3.  To  how   many  degrees  is 

the  sum  of  all  the  angles  of  all  the  triangles  equal? 

4.  To  how  many  right  angles  is  the  sum  of  all  the  angles  of 
the  hexagon  equal? 

5.  Do  all  your   answers  hold   true  for  the  regular   hexagon 
(sides  equal  and  angles  equal)  of  Fig.  181  also? 

6.  In  each  of  these  hexagons  how  many  less  triangles  than 
sides  of  the  uncut  hexagons  are  there? 


FIGURE  181 


CONSTRUCTIVE    GEOMETRY  %\)7 

7.  Draw  an  8-sided  figure  and  find  how  many  triangles  the 
diagonal  cuts  from  any  vertex  would  give. 

8.  How   many  less   triangles   than   sides  wero  given  by  the 
quadrilaterals  of  Fig.  180? 

9.  To  how  many  right  angles  is  the  sum  of  all  the  angles  of 
any  15-sided  figure  equal?  of  an  ^-sided  figure? 

10.  The  angles  of  a  regular  hexagon  are  all  equal.    How  many 
degrees  does  each  contain? 

EXERCISE  8.  —  Draw  a  rectangle  and  cut  it  out.     Cut  it  along 
a  diagonal,  and  denote  the  angles  by  letters,  as  shown  in  Fig.  182. 

1.  Turn  the  lower  right-hand 
triangle  around  and  place  it  on  the 
upper  piece  so  that  e  may  fall  at  «, 
/  at  c,  and  d  at  I.     Can  you  make 
the  parts  fit? 

2.  How  does  a  diagonal  seem  to  divide  a  rectangle? 

3.  Show  the  relative  length  of  sides  m  and  n  by  an  equation  ; 
of  k  and  /;   of  the  angles  a  and  e\   c  and  /;  b  and  d-,  b  +  e  and 
a  +  d. 

4.  From  these  equations  point  out  those  which  show  that  the 
opposite  sides  of  a  rectangle  are  equal. 

5.  Show  from  your  equations  how  the  opposite  angles  compare 
in  size. 

6.  With  parallel  rulers  draw  a  parallelogram  and  cut  it  as  sug- 
gested by  Fig.  182.     Answer  questions  1-5  for  the  parallelogram. 

7.  Write  the  perimeters  of  the  parallelogram  and  of  the  rect- 
angle of  Figs.  180  and  182. 

§189.  Products  of  Sums  and  Differences  of  Lines. 

x  y  1.  What  is  the  area  of  A?  of  B*  of  C? 

ql      A  NOTE.—  The  product  of  a  and  oc  +  y  iswrit- 

_         ten  a(x-\-y)    and  is  read    "a    times   the    sum 


x  +  9  2.  Read  the  equation  ax  +  ay  =  a  (%  +  y) 

and  explain  its  meaning  from  Fig.  183. 
3.  Draw  a  figure  and  show  that 

z). 


A  tviixiM  C/HU 


4.  What  is  the  area  of  each  of  the  parts  of  Fig.  184? 

NOTE.  •  a  -f-  b    times     x-\-y    is    written 


x  +  y 

FIGURE  184 


5.  From  Fig.  184  show  the  meaning  of 
each  product  in  the  equation  (a  -\-b)  (x  +  y) 
-  ax  +  ay  +  bx  +  by.  Also  show  from  Fig. 


184  why  the  equation  is  true. 

6.  What  is  the  area  of  each  part  of  Fig.  185? 

7.  Write  an  equation  showing  how  to  mul- 
tiply  x  +  y  by  itself,  or  how  to  square  x  +  y. 

8.  Point  out  from  Fig.  185  the  meaning  of 

(x  +  y)2  =  x2  +  Zxy  +  y2. 

9.  Point  out  from  Fig.  186  the  meaning  of 

b  (a  —  c)  =  ab  —  be. 


x+y 
FIGURE  185 


I _     •*  -      I  L±U 

EE 


b  b 

FIGURE  186 


a+b  *•    a 

FIGURE  187 


10.  Draw  a  figure  and  show  that 

b  (a  +  d  —  c)  =  ab  +  bd  —  be. 
11.  Show  from  Fig.  187  that 

(a  +  b)(a-b)=az-b\ 
1.-A         12.  Show  from  Fig.  188  that 

(a  -  b)  (a  -  b)  =  a2  -  2ab  -f  < 


a-b 

FIGURE  188 


13.  What  is  the  areaof  the  rectangle  ABCD  (Fig.  189)?  of  S? 

14.  What   is  the  length  of   the  dotted 
part  of  BCt 

15.  What  is  the  area  of  E? 

16.  What  is  the  area  of  the  cross-ruled 
part? 

17.  Write   the  areas   of  the  cross-lined 
figures  below  (Fig.  190). 


FIGURE  189 


STUDY    OF   THE    SUN  S    BAYS 


299 


Call  the  difference  of  the  two  areas  d  in  each  case,  and  answer 
with  an  equation. 

(1)  (2)  (3)  (4) 


FlGUBB  190 

18.  A  lot,  50'  x  175',  is  occupied  by  a  house  which,  with  its 
porch,  covers  a  40'  square.     A  cement  walk  3^'  wide  runs  from 
the  front  line  of  the  lot  to  the  porch,  and  from  the  back  door  of 
the  house  to  the  rear  line  of  the  lot.     The  rest  of  the  lot  is  covered 
with  grass.     How  many  square  feet  of  grass  are  there? 

19.  A  man  cuts  an  18"  strip  of  grass  entirely  around  a  rectan- 
gular lawn,  75'  x  175';  how  many  square  feet  of  grass  does  he  cut? 

20.  He  then  cuts  another  18"  strip  around  just  inside  of  the 
strip  mentioned  in  problem  19;  how  many  square  feet  of  grass 
does  he  cut  the  second  time  round? 

§190.  Distribution  of  the  Sun's  Light  and  Heat. 

Fig.  191  shows  a  simple  instrument,  easily  made  in  a  manual 
training  shop,  with  which  the  varying  slant  of  the  sun's  rays  and 
the  law  of  their  distribution 
may  be  studied.  The  in- 
strument is  called  a  skiam- 
eter. 

DESCRIPTION  OF  THE  IN- 
STRUMENT. —  The  instru- 
ment consists  of  a  smooth, 
straight  stick,  4  in.  (or  10 
cm.)  square  by  12  in.  (or  30 
cm.)  long  (a  square  prism), 
hinged  at  its  lower  end  to  a 
baseboard,  as  shown.  The 
prism  may  be  held  at  any  in- 
clination by  means  of  a  slide  which  works  along  a  slot  and  may  be 
clamped  by  a  thumb-nut  at  the  rear.  An  essential  part  of  the 
apparatus  is  the  brass,  or  paper,  protractor  at  whose  center  a  short 
plumbline  is  so  suspended  as  to  swing  freely  just  in  front  of  its 


FIGURE  191 


300  RATIONAL    GRAMMAR    SCHOOL    ARITHMETIC 

outer  surface.  The  protractor  should  be  tacked  in  such  a  position 
that  the  plumbline  may  swing  exactly  over  the  zero  (0)  when  the 
prism  lies  in  a  level  plane.  Then  as  it  is  raised  the  line  swings 
past  the  successive  graduations  of  the  protractor  so  that  the 
reading  of  the  position  of  the  line  on  the  arc  of  the  protractor 
gives  directly  the  inclination,  or  slant ,  of  the  prism. 

The  plumbline  may  be  made  of  a  thread  and  a  split  bullet. 

If  desired,  instead  of  the  solid  prism,  a  prism-shaped  box  open 
at  both  bottom  and  top  may  be  used.  When  exposed  to  the  sun- 
shine the  rectangular  spot  at  gf  would  then  be  bright.  This  form 
of  the  instrument  is  called  a  helios. 

It  is  also  convenient  to  have  the  baseboard  marked  off  into 
inches  and  quarter  inches  (or  into  centimeters)  along  the  edge,  #/', 
of  the  shadowed,  or  illuminated,  spot.  This  permits  the  direct 
reading  of  the  length  of  the  rectangular  spot.  The  work  and 
description  which  follow  refer  to  the  skiameter.  The  modifica- 
tions to  adapt  them  to  the  helios  are  obvious. 

USE  OF  THE  INSTRUMENT. — On  a  sunny  day  the  baseboard  is 
placed  in  a  level  position  and  turned  so  that  the  prism  may  be 
>•  pointed  directly  toward  the  sun.  The  base- 

A  board  may  be  leveled  by  pouring  a  little  water 

upon  it  and  sliding  thin  wedges  under  one  edge 
or  another  until  the  water  shows  no  tendency 
to  run  in  any  direction.  A  smooth  marble  may 
be  used  instead  of  water.  A  home-made  level 
like  that  shown  in  Fig.  192  is  better  still.  The 

_  „      ,,       mark  on  the  cross-bar  of  the  A,  Fig.  192,  should 

DCd.le  I  — 1±        j,je  ma(je  by  setting  the  A  on  a  level  surface  in 

FIGURE  192  ^he  manual  training  shop  and  marking  it  just 

behind  the  plumbline.  The  A  may  now  be  set  011  the  baseboard 
of  the  skiameter  and  the  baseboard  tipped  by  the  thin  wedges 
until  the  plumbline  swings  freely  over  the  mark. 

Make  such  a  level  from  the  scale  drawing  of  Fig.  192. 

Now  raise  or  lower  the  prism  of  the  skiameter  until  the  shortest 
possible  length,  gf  (Fig.  191),  of  the  rectangular  spot  is  obtained. 
Clamp  the  thumb-nut  behind  the  groove  and  read  on  the  protractor 
the  angle  over  which  the  plumbline  has  swept.  This  angle  is 
the  slant  of  the  sun's  rays.  Measure  and  record  also  the  length, 
#/,  of  the  rectangular  spot. 

The  skiameter  will  enable  us  to  learn  two  things . 

1.  The  way  the  slant  of  the  sun's  rays  changes  during  the  day 
and  from  day  to  day  during  the  year. 

2.  The  law  of  distribution  of  these  rays  over  a  definite  area  of 
the  earth's  surface  at  any  place. 


HOUB 

r3J-jAiN  1    \JE 

RAYS 

ijj^mjrJ-H  \JE 

SPOT 

T  J  A  U  J.  -U 

SPO 

8  a.m.  .  . 

22.7° 

10.37" 

4" 

9  a.m.  .  . 

33.8° 

7.20" 

4" 

10  a.m..  . 

42.0° 

5.98" 

4" 

11  a.m..  . 

48.2° 

5.36" 

4" 

12  a.m..  . 

50.2° 

5.20" 

4" 

1  p.m.  .  . 

48.0° 

5.38" 

4" 

2  p.m..  . 

41.2° 

6.08" 

4" 

3  p.m.  .  . 

32.5° 

7.45" 

4" 

4  p.m.  .  . 

21.0° 

11.16" 

4" 

STUDY    OF   THE    SUN'S   RATS  301 

§191.  Problems  with  the  Skiameter. 

On  a  certain  day  that  promised  to  be  clear,  the  observations 
described  above  were  made  at  8  a.m.,  9  a.m.,  and  so  on  hourly 
to  4  p.m.,  with  results  as  here  tabulated: 

WIDTH  OF        LENGTHS  TO  SCALE 
FOB  SLANT       FOB  SPOT 

2.3"  5.2" 

3.4"  3.6" 

4.2"  3.0" 

4.8"  2.7" 

5.0"  2.6" 

4.8"  2.7" 

4.1"  3.0" 

3.2"  3.7" 

2.1"  5.6" 

The  angles  were  estimated  as  closely  as  possible  from  the  pro- 
tractor to  tenths  of  a  degree.  The  spot  lengths  were  measured 
to  the  nearest  sixteenth  of  an  inch  and  these  measures  were 
reduced  to  hundredths  of  an  inch. 

1.  Why  does  the  width  of  the  spot  remain  always  the  same? 

2.  Draw  a  horizontal,  line  and  a  set  of  9  equally  spaced  parallels, 
all  perpendicular  to  the  horizontal  line  as  shown  in  Fig.  194,  p.  304. 
Call  the  first  parallel  on  the  left  the  8  o'clock  line,  the  second  the 
9  o'clock  line,  and  so  on,  and,  letting  1  in.  represent  10°,  measure 
off  distances  from  the  horizontal  on  the  parallels  to  represent  the 
successive  values  of  the  slant  of  rays  (column  2).    Draw  as  smooth 
a  curve  as  possible  through  the  points  at  the  upper  ends  of  the 
plotted  distances.     This  curve  gives  a  picture  of  the  way  the  slant 
changes  from  hour  to  hour  on  this  day. 

3.  Draw  a  similar  set  of  lines,  or  use  the  same  set,  and  measure 
off  to  the  scale  1 : 2  the  given  lengths  of  the  spot  on  the  corre- 
sponding hour  lines.     Draw  a  smooth  curve  through  the  points  as 
before.     In  what  respects  are  the  two  curves  different? 

4.  Make  a  set  of  these  hourly  measures  some  clear  day  and 
draw  the  curves  from  your  own  measures. 

5.  At  what  hour  is  the  slant  of  the  sun's  rays  greatest?  least? 
G.  At  what  hour  is  the  shadow's  length  the  least?  the  greatest? 


302  RATIONAL    GRAMMAR    SCHOOL   ARITHMETIC 

7.  Make  sets  of  measures  on  days  two  weeks  or  a  month  apart, 
and  plot  them,  preferably  on  the  same  sets  of  parallels.     How 
does  the  shape  of  the  curve  of  slant  change  from  fortnight  to  fort- 
night, or  from  month  to  month. 

Following  is  a  set  of  noon  (12  o'clock)  measures  separated  by 

an    interval  of  30  days    through    the  year    for    latitude  41.9° 
(Chicago) : 

DATE  SLANT  SHADOW  LENGTH 

Jan.     1 26.8°  8.87" 

Jan.   31 32.4°  7.46" 

Mar.    2 42.4°  5.93" 

April   1 54.1°  4.94" 

May     1 64.7°  4.42" 

May  31 71.6°  4.21" 

June  30 73.3°  4.17" 

July  30 68.7°  4.29" 

Aug.  29 59.7°  4.63" 

Sept.  28 48.4°  5.35" 

Oct.   28 37.3°  6.60" 

Nov.  27 29.1°  8.22" 

Dec.  27 26.8°  .            8.87" 

8.  Using  convenient  scales  for  slant  and  for  shadow  lengths 
plot  the  data  of  columns  2  and  3  on  a  set  of  equally-spaced  parallels 
representing  the  dates  of  column  1.     This  curve  shows  the  law 
of  change  of  noon  slant  of  the  sun's  rays  through  the  entire  year. 
Compare  your  curve  with  the  curve  of  Fig.  194,  p.  304. 

9.  On  what  date  is  the  noon  slant  the  greatest?  the  least?     On 
what  date  is  the  shadow's  length  the  least?  the  greatest? 

10.  Look  at  the  curve  and  tell  when  the  slant  is  increasing 
most  rapidly,  least  rapidly.     When  is  it  decreasing  most  rapidly? 

11.  Answer  similar  questions  for  the  lengths  of  the  shadowed 
rectangle. 

12  If  the  inclined  prism  were  not  present  a  prism  of  the  sun's 
rays  4"  square  would  be  spread  over  the  shadowed  rectangle. 
When  would  this  rectangle  be  most  intensely  heated  and  lighted 
by  this  prism  of  rays, — when  the  rectangle  has  the  least  or  the 


STUDY    OF   THE    SUN'S   RAYS 


303 


greatest  area?     Give  a  reason  why  the  earth  is  warmer  in  summer 
than  in  fall,  winter,  or  spring. 

13.  If  the  rectangle  had  an  area  of  18  sq.  in.  at  one  time  and 
36  sq.  in.  at  another,  at  which  time  would  it  be  most  strongly 
heated? 

14.  Recalling  that  the  width  of  the  rectangle  is  always  4"  and 
the  lengths  are  as  given  in  the  third  column  of  the  last  table 
above,  find  all  the  areas  and  the  ratio  of  the  area  for  Jan.  1  to 
the  area  for  each  date  of  the  table. 

15.  How  do  the  areas  of  rectangles,  having  the  same  width, 
change  as  the  lengths  change?     Answer  by  finding  the  ratios  of 
two  areas,  the  ratios  of  their  lengths,  and  comparing  the  ratios. 

16.  The  strength  of  the  sun's  heat  and  light  on  a  given  area 
of  the  earth's  surface  for  any  date  of  the  table  is  how  many  times 
as  great  as  for  Jan.  1? 

17.  Draw  a  set  of  parallels,  or  use  cross-lined  paper,  and  plot, 
to  a  convenient  scale,  the  ratios  found  in  problem  14.     Do  the 
rises  and  falls  in  this  curve  agree  with  the  seasonal  changes  in 
temperature? 

A  still  simpler  device  for  obtaining  more  accurately  the  slant 
of  the  sun's  rays  for  any  time  is  the  one  shown  in  Fig.  193. 

CONSTRUCTION  OF  APPARATUS. — Cut 
and  surface,  from  inch  lumber,  two  boards 
of  dimensions  shown  in  the  cut.  Square 
up  one  edge,  PN,  of  the  square  board 
to  an  accurate  right  angle  with  the  adja- 
cent face  MNPO. 

USE  OF  THE  APPARATUS. — By  the  aid 
of  the  wedges  place  the  rectangular  board 
horizontal  (indicated  by  pouring  a  little 
water  or  placing  a  marble  upon  it),  and 
then  set  the  square  board  edgewise  upon 
it  and  edgewise  to  the  sun  in  such  posi- 
tion as  to  make  the  shadow  at  8  as  narrow 
as  possible.  Stick  a  pin  in  the  face  of  the  vertical  board  at  a 
about  2  inches  from  the  edge  PN.  Stick  a  second  pin  at 
b  so  that  its  shadow  shall  fall  upon  that  of  pin  a.  Turn  the 
board  about  so  that  edge  OP  shall  be  toward  the  sun,  and  make 
the  shadow  S  its  narrowest  again.  Stick  a  third  pin  at  e,  as  the 


FIGURE  193 


304 


RATIONAL   GRAMMAR   SCHOOL   ARITHMETIC 


pin  I  was  placed  formerly.  Through  the  pin-marks  draw  the 
lines  ab  and  «c,  forming  the  angle  bac.  Bisect  (see  Problem 
II,  p.  186)  the  angle  bac  with  the  line  az  and  measure  baz  with  a 
protractor.  This  gives  the  angle  between  the  sun  and  the  zenith 
(the  point  on  the  sky  directly  overhead).  Point  to  the  zenith. 
Move  the  extended  arm  downward  until  it  points  toward  the 
horizon  (the  earth  and  sky  line).  Through  how  many  degrees 
of  angle  does  your  arm  move? 

The  slant  of  the  sun's  rays  is  the  difference  between  baz  (or 
caz)  and  90°.  Why? 

One  half  of  cad  would  be  the  slant  directly.     Why? 

18.  Solve  problems  1-7  with  this  apparatus. 

19.  Measure  and  plot  the  noon  slant  from  month  to  month 
during  the  school  year. 

20.  With  a  carefully  drawn  curve  made  hourly  from  measures 
(Fig.  194)  with  the  boards  and  pins,  the  time  of  apparent  noon 

(sun  noon)  may  be 
found.  Draw  a  line 
as  AB  parallel  to 
EF  and  cutting 
the  curve  at  two 
points,  A  and  B. 
With  compasses 
bisect  A  B  with  the 
perpendicular  ON. 
The  highest  point 
of  the  curve  being 
the  noon  point, 
noon  occurs  such 
a  fractional  part  of  an  hour  before  12  as  the  distance  JV-12 
is  of  the  distance  11-12.  From  a  curve  of  your  own,  measure 
these  distances  and  compute  the  clock  time  of  apparent 
noon. 

From  problem  15  it  is  clear  that  instead  of  comparing  the 
areas  of  the  rectangles  we  may  compare  their  lengths.  The 
shorter  the  rectangle  the  greater  the  amount  of  heat  and  light  on 
any  given  surface  at  a  place. 

In  latitude  50°  north  (meaning  50°  north  of  the  equator)  the 


/ 

\ 

M 

^T 

-^ 

A 

/ 

10 
cvi 

k>^ 

^ 

\ 

M 

\ 

\ 

CVJ 

\ 

cvi 

w 

(VI 

00 

o 

in 

5 

i 

E 

N 

F 

8      9      10     II      IE      1      2      3     4 

FIGURE  194 

STUDY   OF  THE   SUN'S   RAYS  305 

lengths  of  the  rectangles  would  be  found  to  be  as  in  this  table  for 
the  dates  given : 

DATE  LENGTH  HEAT                        DATE  LENGTH          HEAT 

Jan.     1..  8.84"  1  heat  unit         July     1..  4.18" 

Jan.  15..  8.33"  1.06                    July   15.  .  4.21" 

Feb.     1..  7.40"  1.19                    Aug.     1..  4.31" 

Feb.  15  ..  6.65"  Aug.  15..  4.44" 

Mar.    1..  5.97"  Sept.    1..  4.69" 

Mar.  15  . .  5.39"  Sept.  15. .  4.98" 

Apr.    1  ..  5.01"  Oct.      1..  5.47" 

Apr.  15..  4.52"  Oct.    15..  6.00" 

May    1  ..  4.42"  Nov.     1..  6.82" 

May  15..  4.29"  Nov.  15..  7.61" 

June   1  ..  4.21"  Dec.     1..  8.44" 

JunelS..  4.17"  Dec.   15..  8.87" 

21.  Using  a  scale  of  1 :  2,  plot  the  lengths  of  the  rectangles  for 
the  24  dates  given  and  draw  a  smooth  curve  through  the  points. 
When  does  the  rectangle  shorten  most  rapidly?  least  rapidly? 

22.  Suppose  a  given  surface  (say  1  sq.  ft.)  is  heated  by  1  heat 
unit  on  Jan.  1.     Compute  the  number  of  heat  units  the  same  sur- 
face receives  on  Jan.  15;  on  Feb.  1;  on  the  remaining  dates. 

SOLUTION. — Call  the  required  number  of  heat  units,  JET. 

TT  Q     QJ 

Then,  -^  =  ~  ~  =  1.06  (to  2  decimals). 

1  o.Ou 

For  Feb.  1,      -^  =  ~  =  1.19  (tp  2  decimals). 

23.  Using  the  scale  V  to  1  heat  unit,  plot  for  the  given  dates 
the  computed  values  to  be  filled  into  the  columns  in  the  above 
table.     Do  the  rise  and  fall  in  the  curve  agree  with  the  seasonal 
changes  of  heat? 

24.  How  does  this  curve  (which  belongs  to  the  latitude  of  Van- 
couver,   Winnipeg,    Newfoundland,    Land's    End,    and    Prague) 
compare  with  the  corresponding  curve  for  Chicago?     (Use  the 
values  to  be  filled  into  the  heat  columns  of  the  table  of  problem 
20,  above.) 


306  RATIONAL   GRAMMAR    SCHOOL   ARITHMETIC 

§192.  Longitude  and  Time. 

The  earth  may  be  regarded  as  an  immense  sphere  turning,  like* 
a  top,  round  one  of  its  diameters  as  an  axis  from  west  to  east 
once  every  24  hr.  This  carries  the  surface  of  the  earth  and 
all  objects  fixed  upon  it  (as  a  schoolhouse)  round  through  360° 
in  24  hours. 

Measuring  one  complete  turn  (rotation)  of  the  earth  in  degrees, 
it  may  be  said  to  be  equivalent  to  360°;  measuring  it  in  time,  it 
may  be  said  to  be  equivalent  to  24  hours. 

This  gives  us  the  following  tables  of  equivalent  measures : 

360°  correspond  to  24  hr. ; 

1°  corresponds  to  ^  of  24  hr.  =  ^  hr.     =  4  min.  of  time; 
1'  corresponds  to  FV  of  4  min.   =  T^  min.  =  4  sec.  of  time ; 
V  corresponds  to  ^ff  of  4  sec.     =  y1-  sec.  of  time. 

24  hr.  correspond  to  360°  ; 
1  hr.  corresponds  to  15°; 

1  min.  of  time  corresponds  to  -fa  of  15°  =  t°  =  15'; 
1  sec.  of  time  corresponds  to  fa  of  15'      =  J'  =  15". 

All  objects  seen  on  the  sky,  as  the  sun,  may  be  regarded  as 
stationary,  while  the  turning  of  the  earth  carries  us  past  them 
from  the  west  toward  the  east.  This  makes  the  sun  appear  to 
rise  in  the  east,  move  over,  and  set  in  the  west. 

1.  Will  the  sun  pass  over  eastern  or  western  places  earlier? 
Which  places  have  later  local*  times,  those  over  which  the  sun 
passes  earlier,  or  later?     Which  places  have  earlier  local  times, 
eastern  places,  or  western  places? 

2.  What  time  is  it  at  a  place  45°  west  of  Washington  when 
it  is  10  o'clock  at  Washington?     At  the  same  instant,  what  time 
is  it  at  a  place  30°  east  of  Washington? 

DEFINITION. — Longitude  is  the  distance  in  degrees,  minutes  and 
seconds  (of  arc)  due  eastward  or  westward  from  a  chosen  meridian,  called 
the  prime  meridian.  Astronomers  and  navigators  have  agreed  that  the 
prime  meridian  shall  be  the  meridian  of  the  Royal  Observatory  at  Green- 
wich, England. 

3.  The  difference  between  the  local  times  of  two  places  is  4hr. 
3  min. ;  what  is  the  difference  in  longitude  between  them? 

*  Local  sun  time  is  obtained  by  setting  timepieces  at  XII  as  the  sun  crosses  the 
meridian. 


LONGITUDE    AND    TIME  307 

4.  The  difference  of  longitude  between  two  places  is  105°  45'; 
what  is  the  difference  of  their  local  times? 

5.  It  is  4:  50  (4  hr.  50  min.)  p.m.  at  a  certain  place  and  1 :  48 
p.m.  at  another;  which  place  is  east  of  the  other  and  what  is  the 
difference  of  their  longitudes? 

6.  Two  men  met  in  Chicago  with  their  watches  keeping  the 
correct  local  times  of  the  places  whence  they  came.    On  comparing 
their  times  one  watch  showed  10:47  p.m.,  and  the  other  3:18 
p.m.,  when  it  was  9:  26  p.m.  in  Chicago.     From  which  direction 
did  each  man  come? 

7.  If  the  watches  (problem  6)  were  keeping  the  correct  local 
times  of  their  places,  what  were    the   differences   of  longitude 
between  the  places  and  Chicago? 

8.  Explain  how  it  happened  that  the  announcement  of  Queen 
Victoria's  death  was  read  in  the  Chicago  dailies  at  an  earlier  hour 
than  that  borne  by  the  announcement  itself. 

9.  How  may  it  happen  that  a  cable  message  sent  Wednesday 
forenoon  may  be  received  at  a  remote  place  Tuesday? 

10.  The  local  times  of  two  ships  at  sea  differ  by  4  hr.  18  min. 
15.6  sec. ;  what  is  the  difference  of  their  longitudes? 

11.  One  ship  is  in  longitude  186°  40'  12"  west  and  another  is 
in  longitude  20°  16'  48"  west;  what  is  the  difference  of  their  longi- 
tudes? of  their  times? 

12.  The  longitude  of  one  ship  is  3°  28'  10"  west  and  that  of 
another  is  18'  38"  east.    What  is  the  difference  of  their  longitudes? 
of  their  times? 

13.  The  longitude  of  the  Observatory  of  Madrid,  Spain,  is 
3°  41'  17"  west  and  that  of  the  Berlin  Observatory  is  13°  23'  43" 
east.      What  is   the    difference  of    their    longitudes?    of    their 
times? 

14.  The  longitudes  of  the  Cambridge  (Eng.)  and  of  the  Paris 
Observatories  are  5'  41.25"  east   and  2°  21'  14.55"  east  respec- 
tively.    What  are  the  differences  of  their  longitudes?    of  their 
times? 

15.  When  the  sun  is  on  your  meridian,  that  is,  when  it  is  noon 
at  your  place,  at  what  places  on  the  earth  is  it  forenoon?  after- 
noon? night?  midnight? 


308  RATIONAL    GRAMMAR    SCHOOL   ARITHMETIC 

The  longitudes  from  Greenwich,  Eng.,  of  places  are  also 
often  given  (as  below)  in  hours,  minutes  and  seconds  of  time. 
The  plus  (+)  sign  means  that  the  place  is  west  of  the  meridian 
of  Greenwich,  and  the  minus  (— ),  that  the  place  is  east  of  this 
meridian. 

•pT  .  __  LONGITUDE  FROM    «_  .„„  LONGITUDE  FROM 

GREENWICH  GREENWICH 

H.     M.          S.  H.    M.  S. 

Albany +  4  54  59.99  Denver +6  59  47.63 

Algiers -0  12  08.55  Edinburgh +0  12  44.2 

Allegheny,  Penn  ..  +  5  20  02.93  Glasgow +0  17  10.55 

Ann  Arbor,  Mich.  +  5  34  55.19  Madison,  Wis +5  67  37.93 

Berkeley,  Calif.  .  .  +  8  09  02.72  Madrid -1-0  14  45.12 

Berlin -0  53  34.85  Mexico +6  36  26.73 

Bombay -4  51  15.74  New  York.. ......  +4  55  53.64 

Cambridge  (Eng.)  -0  00  22.75  Paris -0  09  20.97 

Cambridge  (Mass.) +  4  44  31.05  Philadelphia +5  00  38.51 

Cape  of  Good  Hope -1  13  54.76  St.  Petersburg  ...- 2  01  13.46 

Chicago +5  50  26.84  Washington +5  08  15.78 

16.  Give  from  the  table  the  difference  of  local  times  of  Green- 
wich, England,  and  of   each  of   the  following   places  and  state 
whether  the  time  is  earlier  or  later  than  Greenwich  time :  Albany ; 
Algiers;  Ann    Arbor;    Berlin;   Bombay;    Chicago;    New  York; 
Paris;  St.  Petersburg. 

17.  What  is  the  longitude  in  degrees  (°),  minutes  (')  and  sec- 
onds (")  of  arc  of  Albany?   of  Algiers?    of  Cambridge   (Mass.)? 
of  Washington? 

18.  Which  of  each  of  these  pairs  of  places  has  the  earlier  local 
time  and  how  much  earlier  is  this  time : 

Albany  and  Allegheny?  Chicago  and  Paris? 

Ann  Arbor  and  Berkeley?  Edinburgh  and  Madison? 

Ann  Arbor  and  Bombay?  Cape  of  Good  Hope  and  Bombay? 

Chicago  and  Cambridge, Mass.?  Paris  and  Bombay? 

19.  Give  the  differences  of  longitude  in  each  case  of  problem 
18. 

20.  Solve  other  similar  problems  on  the  table. 


LONGITUDE    AND   TIME 


309 


§193.  Standard  Time. 

For  convenience  of  railway  traffic  a  uniform  system  of  time- 
keeping, known  as  Standard  Time,  was  agreed  upon  in  1883  by 
the  principal  railroad  companies  of  North  America.  It  was 
decided  that  places  within  a  belt  of  15°  extending  (roughly) 
7£°  on  each  side  of  the  75th  meridian  west  of  Greenwich  should 
use  the  time  of  the  75th  meridian.  All  places  in  similar  belts 
extending  about  7^°  on  each  side  of  the  90th,  of  the  105th,  and 
of  the  120th  meridian  should  take  the  times  of  those  meridians 
respectively,  and  hence  should  have  times  just  1  hr.,  2  hr.,  and 
3  hr.  earlier  (less)  than  the  time  of  the  75th  meridian  time  belt. 
This  system  has  now  been  generally  adopted  by  most  civilized  coun- 
tries. In  practice,  however,  the  dividing  lines  of  the  time  belts 
are  irregular  lines  running  through  railroad  terminals.  The  fol- 
lowing map  shows  the  time  belts  and  the  names  used  to  dis- 
tinguish the  times  of  the  several  belts  which  cover  continental 
United  States.  In  this  system  the  times  at  all  places  in  the 
United  States  differ  only  by  whole  hours. 

PACIFIC  TIME          MOUNTAIN  TIME          CENTRAL  TIME          EASTERN  TIME 
120°  105°  90°  75° 


FIGURE  195 


1.  When  it  is  8  o'clock  a.m.  (Standard  Time)  at  Chicago,  what 
is  the  time  at  each  of  the  following  places :  New  York?  Pittsburg? 


310 


RATIONAL    GRAMMAR    SCHOOL   ARITHMETIC 


St.  Louis?  Kansas  City?   Denver?  Spokane?  San  Francisco?    (See 
the  map,  Fig.  195.) 

2.  When  it  is  9  hr.  10  min.  45  sec.  p.m.  at  Omaha,  what  is 
the  time  at  New  Orleans?    Philadelphia?   Buffalo?  Washington? 
Austin  (Tex.)?  Boise  City?  Los  Angeles?  El  Paso?  Salt  Lake  City? 
(Refer  to  your  Geography.) 

3.  Answer  the  questions  of  problem  2  for  2  :  15  p.m.  at  St.  Paul  ; 
for  3:  25  a.m.  at  Dodge  City,  Kan. 

4.  Answer  for  other  places  marked  on  the  map  such  questions 
as  are  asked  in  problems  2  and  3  for  Omaha,  for  St.  Paul,  and  for 
Dodge  City. 

The  circle  SqNr  (Fig.  196)  represents  the  earth  and  the  curved 
lines  represent  the  hourly  meridians  running  from  the  equator, 

<2T,  and  converging  toward  the 
poles  S  and  N.  If  0  represents 
F  some  place  in  the  northern  hemi- 
sphere, SqNr  is  the  meridian  of 
the  place  and  BQPR  may  be  im- 
agined to  represent  the  meridian 
of  the  sky  (hour  circle  of  the  sun)  , 
on  which  the  sun,  T,  is  situated. 
The  hour  circle  of  the  sun  in  the 
apparatus  is  held  in  place  at  A.  If 
now  the  crank  F  is  turned  so  as 
to  carry  q  forward  and  downward 
through  E  to  r,  the  sun's  meridian 
standing  stationary,  the  motions  which  cause  the  differences  of 
time  and  the  changes  of  day  and  night  may  be  understood,  by 
recalling  that  only  the  half  of  the  globe  which  is  turned  to  the 
sun  is  light  (in  day)  .  SGN  denotes  the  prime  meridian. 

5.  As  the  globe  is  standing  in  Fig.  196,  what  time  is  it  at  0? 
at  places  on  the  105th  meridian?  on  the  60th  meridian?  on  the 
15th  meridian  east*  of  Greenwich?  on  the  45th  east? 

6.  If  the  meridian  8GN  is  continued  around  on  the  other 
side  of  the  globe,  what  will  its  number  be? 

7.  Remembering  that  any  place  on  the  earth,  as  0,  turns  com- 
pletely round  through  360°  in  24  hr.,  how  long  does  it  require 


csru 


*  Remember  that  east  is  the  direction  toward  which  the  globe  is  turning,  that  is  east- 
ward means  from  E  toward  G  through  r  and  around  to  E  again. 


LONGITUDE    AND   TIME 


311 


the  space  between  any  two  adjacent  meridians  shown  in  Fig.  196 
to  pass  under  the  sun's  hour  circle? 

8.  If  a  man  should  start  from  some  place  on  the  prime 
meridian  (say  London)  on  Friday  noon  and  move  westward  just 
as  fast  as  the  globe  turns  eastward,  what  time  (by  the  sun)  would 
it  be  to  him  during  his  journey  all  the  way  round  the  globe? 
What  hour  and  day  would  a  Londoner  call  it  when  the  traveler 
returned  24  hr.  later? 

The  problem  raises  the  question,  "Where  should  the  traveler 
have  changed  his  date  so  that  his  date  might  agree  with  that  of 
his  starting  place  when  he  returns?"  The  answer  is,  "It  has 
been  agreed  that  the  date  should  change  at  the  180th  meridian." 
When  vessels  cross  this  meridian  from  the  east  toward  the  west 
they  add  a  day  to  their  reckoning.  If  they  cross  at  noon  on 
Friday,  Friday  noon  instantly  becomes  Saturday  noon.  Crossing 
from  the  west  toward  the  east,  they  repeat  a  day.  In  the  case 
mentioned,  Friday  would  "be  done  over  again."  The  180th 
meridian  is  for  this  reason  called  the  Date  Line.  Trace  it 
round  the  earth  in  your  Geography. 


MENSURATION 


§194.  Roofing  and  Brick  Work. 

1.  Give  the  rule  for  computing  the  area  of  a  square  from  its 
measured  sides  ;  a  rectangle  ;  a  parallelogram  ;  a  triangle  ;  a  trape- 
zoid.  (See  pp.  114-118  and  189.) 

DEFINITION.  —  A  square  of  roofing  means  a  10'  square  of  roof  surface  or 
100  sq.  ft.  A  shingle  is  said  to  be  laid  4",  4£",  or  5"  to  the  weather 
when  the  lower  end  of  each  course  of  shingles  on  the  roof  extends  4",  4|", 
or  5"  below  the  course  next  above  it. 


2.  Draw  two  perpendicular  center 
lines,  and  with  the  aid  of  the  dimen- 
sions given  in  the  left  part  of  Fig.  197, 
complete  an  enlarged  drawing  to  a  con- 
venient scale,  of  the  development  of  the 
roof,  (shown  on  the  right.) 


FIGURE  197 


312 


RATIONAL    GRAMMAR    SCHOOL   ARITHMETIC 


3.  If  1000  shingles  laid  4"  to  the  weather  cover  a  square  of 
roofing,  how  many  shingles  will  be  needed  to  cover  a  square,  if 
laid  5"  to  the  weather?  4£"  to  the  weather?    3"  to  the  weather? 

a*"? 

4.  The  dimensions  on  the  development  (Fig.  197)  being  in 
feet,  find  the  cost  of  the  shingles,  at  $1.10  a  bunch  of  250,  needed 
to  cover  the  four  sides  of  the  deck  roof   (Fig.  197).     Find  the 


FIGURE  198 

cost  of  enough  tin  to  cover  the  12'  x  16'  flat  deck  at  15^-  a  20"x  28" 
sheet,  the  long  sides  of  the  sheets  being  laid  parallel  to  the  long 
side  of  the  deck,  and  allowing  10%  loss  for  joints  and  overlap  at 
edges. 

5.  When  shingles  are  laid  4"  to  the  weather,  1000  shingles  are 
estimated  to  cover  a  square.  Find  the  number  of  shingles  laid 
4"  to  the  weather,  needed  to  cover  the  roof,  including  the 
gables,  dimensions  being  as  given  in  Fig.  198,  and  supposing  the 
other  side  and  the  end  to  require  the  same  number  of  shingles  as 
do  the  side  and  the  end  shown  in  the  figure.  (The  upper  part 


MENSUKATION 


313 


of  the  figure  represents  the  house  facing  toward  the  right.  The 
lower  part  represents  the  front  end  of  the  house.)  Find  the  cost 
of  the  shingles  at  90^  a  bunch  of  250. 

6.  Ten  years  after  building  this  house  (Fig.  198)  a  new  roof 
was  put  on  it.  It  cost  $1.25  per  M.  to  remove  the  old  shingles, 
$3  per  M.  to  dip  the  new  ones,  and  $2.75  per  M.  for  the  labor  of 

putting  on  the 
shingles.  How 
much  did  it  cost 
to  remove  the 
old  shingles 
and  to  re-roof 
with  dipped 
shingles? 

7.  The  roofs 
of  my  neigh- 
bor's house  and 
porches  are  as 
shown  in  Fig. 
199.  It  is  covered 
with  slates,  240 

to  the  square  (100  sq.  ft.)  of  roof.  The 
towers  are  octagonal  (8-sided),  the  sides 
all  being  equal  triangles.  The  dimensions 
are  given  in  feet.  Find  the  number  of 
slates  needed  for  the  front  of  the  roof  as 
shown. 

8.  Draw  the  development  of  the  tower  as  shown  in  Fig.  199. 

9.  If  it  takes  14  brick  per  square  foot  of  outside  surface  to  lay 
a   2 -brick   wall,    how   many   brick  will    be 

needed  to  lay  the  side  wall  of  the  build- 
ing shown  in  Fig.  200,  deducting  for  5 
windows  each  3^'  x  6-^'  and  for  5  windows 
each  3f  x  9f. 

10.  Make  other  similar  problems   from 
your  own  measurements  or  from  dimensions 

obtained  from  an  architect  or  builder.  FIGURE  200 


FIGURE  199 


314 


RATIONAL    GRAMMAR    SCHOOL    ARITHMETIC 


§195,  Land  Measure. 

Keview  §83,  pp.  125  and  126. 

The  law  requires  land  to  be  marked  out  or  surveyed  in  divisions 
of  the  form  of  squares  and  rectangles.  In  the  western  states  the 
land  has  been  surveyed  in  accordance  with  this  law. 

To  mark  out  the  largest  squares,  north  and  south  lines,  called 
meridians,  are  first  run  24  mi.  apart  and  marked  with  corner- 
stones, or  by  trees,  or  other  permanent  objects.  East  and  west  lines, 
called  base  lines,  are  then  run  at  right  angles  to  these  meridians 
at  distances  24  mi.  apart.  This  would  divide  the  land  up  into 
24-mi.  squares  were  it  not  for  the  convergence  of  the  meridians 
toward  the  poles  of  the  earth.  Notice  this  on  a  map  in  your 
Geography  and  on  the  map  of  Fig.  195. 

Each  24-mi.  tract  is  then  divided  into  16  nearly  equal  squares, 

called  townships,  by  running  north  and  south,  and  east  and  west 

lines  through  the  quarter  points  of  the  sides  of  the  large  tract. 

How  long  is  a  township?  how  wide?  how  many  square  miles 

does  it  contain? 

Certain  meridians,  called  principal  meridians,  are  run  with 
great  care,  and  these  principal  meridians  govern  the  surveys  of 
lands  lying  along  them  for  considerable  distances  both  toward 
the  east  and  toward  the  west.  The  tiers,  or  rows,  of  townships 

running  north  and  south  along 
the  principal  meridians  are 
called  ranges.  The  first  tier  on 
the  east  is  called  range  No.  1 
east,  and  is  written  K1E;  the 
second  range  is  No.  2  east, 
written  I12E,  and  so  on. 

Point  out  on  the  drawing, 
Fig.  201,  R1W;  E2W;  R3W; 
R2E;  E4W. 

Certain  base  lines  are  run 
with  great  care  and  are  called 
standard  base  lines.  The  rows 
of  townships  running  east  and 


*        *        *        £    . 

•*•         tf>        «\j         — 
oe.te.cc.ac. 
Stands 

Ul          Ul 

—               (M 

CC                IX 

rd  Line 

1 

4 

4 

4 

4 

4 

^ 

3 

3 

(*> 

Meridia 

3 

3 

Z 

* 

Z 

| 

2 

Z 

1 

' 

I 
Ba 

s 

1 

se  Lin 

1 

3 

1 

I 

' 

1 

1 

i 

1 

west  are  numbered  with  refer- 
FiGUBE2i)i  ence    t0   thggg   standard    base 

lines.  A  township  in  the  first  row  north  of  a  base  line  is  town- 
ship No.  1  north,  and  is  written  TIN;  one  in  the  second  row 
south  is  called  township  No.  2  south,  written  T2S,  and  so  on. 

Interpret  the  following  symbols  and  point  out  on  the  drawing, 
Fig.  201,  the  townships  indicated:  T3N;  T4N;  T1S;  T2N. 


MENSURATION 


315 


A  township  is  identified  by  giving  its  number  and  range  from 
some  standard  base  and  principal  meridian. 

Point  out  these  townships  on  the  drawing,   Fig.  201 :    TIN, 
R2W;    T3N,    E1W;   T4N,  K2E;  TlS,   K3W. 

The  law  also  requires 
townships  to  be  sub- 
divided into  smaller 
squares,  called  sections. 
Sections  are  numbered, 
beginning  at  the  north- 
east corner  and  run- 
ning toward  the  west 
to  6,  then  7  is  just 
south  of  6,  and  so  on, 
as  in  Fig.  202. 

Sections  are  then  sub- 
divided into  quarters  (see 
section  16),  half-quarters 
(see  sections  13  and  26), 
and  quarter-quarters  (see 
section  29). 

1.  Eef erring  to  Fig. 
202,  read  and  write  the 
descriptions  of  the  divi- 
sions of  section  16;  of  section  26;  of  13;  of  29. 

Whatever  deviations  there  may  be  from  exactly  640  A.,  in  the 
sections  of  any  township,  due  to  convergence  of  meridians  or  other 

TownshD  Lin*  causes>  are  required  by  law 

to  be  added  to,  or  sub- 
tracted from,  the  north  and 
west  rows  of  half -sections. 
These  tracts  are  then  not 
called  fractional  sections, 
but  are  called  lots,  and  are 
numbered  in  regular  order 
as  shown  in  Fig.  202. 

A  section  then  always 
means  exactly  640  A.  Any 
fractional  part  of  a  section 
means  the  corresponding 
fractional  part  of  640  A. 
The  lot,  on  the  contrary, 
must  always  be  measured 
FIGURE  203  before  its  area  is  known. 


Lot 

Ib 

Lot  15 
321  A 

Lot  i4r 

JLot  13 

Lot  IZ 

Lot  II 
318  A 

$£ 

6 

5 

4 

3 

Z 

1 

00 

7 

8 

9 

10 

II 

IZ 

Cj"^ 

18 

17 

i 

15 

14 

i  -\ 

SE4 

i  j 

p 

s 

19 

ZO 

Zl 

ZZ 

Z3 

Z4 

>ro 

30 

Z8 

Z7 

Z5 

a 

c\ 

?  f\ 

v  — 

J 

t.  O 

ro 

31 

3Z 

33 

34 

35 

36 

FIGURE  202 


L48 

L47 

L   46 

L  45        1  L44 

L43 

38 

J.CH 

41 
URCH 

79  A 
W.  BROWN 

79  A            39A 
J.  BRADEN    P 

38A 

L49 

1— 

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4-IA 

40 

80 

80         §40 

240 

4 

°C        f 

M  .  EVANS 

H.  BRADEN  }g 

L50 

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j_ 

82  A 

§ 

160 

160 

leo 

Sao- 

\ 

Sect 

on  Line 

i 

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i 

LSI 

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3? 

8IA 

80. 

160 

160 

80 

c_ 

80 

S.PERRY 

m 

L52 

C.  PAI 

KS 

t 

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1 

8IA 

80 

160 

160 

?80 

o 
•<80 

H.BE 

VENS 

316  RATIONAL    GRAMMAR    SCHOOL    ARITHMETIC 

2.  Following  is  an  assessment  list  of  farm  property.     Fill  the 
blanks  from  Fig.  203 : 

OWNER                                      DESCRIPTION  No.  ACRES 

H.  Peabody E£  SEJ  Sec.  8  

H.  Peabody EJ  NEJ  Sec.  8          

J.  White E£  SEi  Sec.  5 

J.  James SEJ  NEJ  Sec.  5        

J.  James Lot  43  .... 

0.  Gibson Lot  44  

0.  Gibson SWJ  NEJ  Sec.  5  .... 

3.  Make  out  an  assessment  list  for  all  owners  in  Sec.  6. 

4.  Make  and  solve  a  similar  problem  for  owners  in  Sec.  7. 

5.  Correct  the  mistakes  in  this  erroneous  assessment  list: 

OWNER  DESCRIPTION  No.  ACRES 

S.  Perry Wi  NWJ  Sec.  7  81 

J.  Hay WJ-  SWJ  Sec.  6  82 

J.  Hay Wi  SWi  Sec.  7  80 

H.  Ochiltree SWJ  NWJ  Sec.  6  41 

H.  Ochiltree SEJ  NEJ  Sec.  6  40 

H.  Ochiltree EJ  SEJ  Sec.  6  80 

J.  Church NWi  NWJ  Sec.  6  38 

J.  Church NEi  NWi  Sec.  6  41 

§196.  Volumes. 

DEFINITION. — The  volume  of  any  figure  is  the  number  of  cubical  units 
within  its  bounding  surfaces. 

1.  Give  the   rule  for  finding  the  volume  of   a   square  prism 
(called  also  a  rectangular  parallelepiped) .    (See  pp.  118,  119.) 

2.  Give  a  rule  for  finding  the  volume  of  an  oblique  parallele- 
piped (Fig.    148,  p.   281)  having  the  same  base  and   the  same 
altitude  as  a  given   rectangular  parallelepiped.     (See  Figs.   150 
and  151,  pp.  281  and  282.) 

3.  How  would  you  find  the  volume  of  a  hollow  beam  12  ft. 
long,  having  a  cross  section  like  (1)  Fig.  75,  p.  138?  (2)?  (3)? 
(5)?  (6)?  (4)?  (9)? 


MENSURATION  317 

4.  Giv-e  the  volumes  of  the  beams  of  problem  3,  if  the  length 
of  the  beam  in  each  case  is  I  feet. 

5.  A  model  of  a  square  prism  (like  Fig.  145) 


having  a  base  2  in.  square  and  an  altitude  of  8  in. 
will  contain  how  many  cubic  inches  of  sand? 

6.  The  model  of  a  right  circular  cylinder, 
made  as  shown  in  the  scale  drawing  of  Fig.  204 
and  pasted  along  the  flap,  DF,  and  around  one 
end  with  a  strip  of  paper,  was  filled  with  sand 


Scale  1:8 


and  poured  into  the  empty  model  of  the  square  G 

prism  of  problem  5.      It  filled  the  square  prism 

a  little  more  than  f  full.     About  how  many  cubic         FIGURE  204 

inches  are  there  in  the  model  of  the  cylinder? 

7.  It  is  shown  in  geometry  that  the  volume  of  any  right  cir- 
cular cylinder  is  .7854  (= -^)  times  the  volume  of  a  square  prism 
of   the   same  altitude  and  having  for   one   side  of   its   base  the 
diameter  of   the  cylinder.     Find   the  volumes  of   these  circular 
cylinders : 

(1)  Diameter    4",  altitude  6"        (4)  Diameter  18£",  altitude  8" 

(2)  "  7"  7"        (5)         "  6'  "      10.5' 

(3)  "         10"         "         8J"       (G)         "  2r  "        a 

8.  How  long  is  AB?  How  many  square  inches  in  the  rectangle 
ADFG  (Fig.  204)?     How  many  square  inches  are  there  in  the  en- 
tire outside  surface  of  the  cylinder? 

9.  The    inside   diameter  of  the  cylindrical  water  tank  of  a 
street  sprinkler  is  3.0  ft.,  and  its  length  is  10  ft. ;  how  many  liquid 
gallons  does  the  tank  hold? 

10.  The  tank  of  the  sprinkler  is  filled  through  a  hose  of  2-J-" 
inside  diameter  from  a  hydrant  from  which  the  water  flows  at  the 
rate  of  300  linear  feet  per  minute.  •   How  long  will  it  take  to  fill 
the  tank? 

SUGGESTION. — In  1  min.  a  cylinder  of  water  as  large  as  the  inside  of 
the  hose  and  300 '  long  flows  into  the  tank. 

11.  Find  the  weight  of  an  iron  rod  £"  in  diameter  and  24'  long, 
if  iron  weighs  450  Ib.  per  cubic  foot? 


318  RATIONAL    GRAMMAR    SCHOOL    ARITHMETIC 

12.  How  many  cubic  inches  of  air  are  there  in  the  hollow  tire 
of  a  bicycle  wheel  28"  in  diameter  (from  center  line  to  center  line 
of  tubes),  the  hollow  having  a  diameter  of  If"? 

13.  How  many  cubic  inches  of  rubber  are  there  in  the  hollow 
tire  of  a  32"  automobile  wheel  if  the  inside  diameter  of  the  cylin- 
drical tube  is  3"  and  the  outside  diameter  is  4  inches? 

14.  Fig.  205  is  a  scale  drawing  of  a  pattern 
for  the  paper  model  of  a  cone  whose  base  is  to 
be  a  circle  2"  in  diameter  and  whose  sloping 
side  from  the  apex  to  the  base  is  to  be  8".  The 
arc  A  LIB  is  just  as  long  as  the  circumference 
of  the  base.  Compute  the  length  of  the  cir- 
cumference of  a  circle  whose  radius  is  2",  and 
of  another  whose  radius  is  8",  and  find  the 
ratio  of  the  first  to  the  second. 
FIGURE  205  15>  Tfthat  part  of  the  whole  circumference 

(with  center  C)  is  the  arc  AHB?     What  part  of  360°  is  the  angle 

ACB? 

16.  Make  a  right  angle,  as  HCX,  and  bisect  it  (See  Problem  II, 
p.  186).     CD  is  the  bisector.     Bisect  angle  HOD  and  obtain  CB. 

17.  With  C  as  center  and  with  a  convenient  radius,  as  CE, 
draw  the  indefinite  arc  PNME.    Put  the  pin-foot  on  G  and  mark 
an  arc  across  arc  GP  at  N.    Draw  ON  and  prolong  it.  This  makes 
angle  GCN  equal  to  angle  GCM.     Now  with  (7  as  a  center  and 
with  8"  as  a  radius,  draw  the  arc  AHB.     Prolong   CH,  make 
HO  =  1",  and  draw  the  lower  circle.     Provide  the  flap  and  paste 
up  the  model  of  the  cone. 

If  such  a  model  is  carefully  made  and  the  height  is  measured 
and  if  the  model  of  a  cylinder  has  the  same  height  and  the  same 
base,  the  model  of  the  cone  will  be  found  to  hold  ^  as  much  sand 
as  does  the  model  of  the  cylinder.  It  is  proved  in  geometry  that 
the  volume  of  any  cone  equals  ^  of  the  volume  of  a  cylinder  having 
an  equal  base  and  an  equal  altitude. 

18.  Find  the  volumes  of  these  circular  cones: 

(1)  Diameter  of  base  3",    altitude,  7" 

(2)  "  "    6"  "       20" 

(3)  "  "     6.8"        "         2.25' 

(4)  "  "     3.95"  10.25" 


MENSURATION  319 

19.  How  many  cubic  inches  of  water  are  there  in  a  funnel- 
shaped  vessel,  if  the  water  is  4"  deep  and  the  diameter  of  the 
surface  of  the  water  is  4.5"?     (See  Fig.  206). 

20.  How  many  cubic  inches  of  water  will  ifc 
take  to  fill  the  same  vessel  to  twice  the  depth,  or 
8"?     (See  Fig.  206.) 

21.  How  many  cubic  inches  had  to  be  poured  yt    * 
into  the  vessel  to  fill  it  to  8"  depth  if  the  depth  of 

the  water  was  4"  at  the  beginning?  FIGURE  206 

22.  A  mountain  peak  has  the  shape  of  a  circular  cone  whose 
altitude  is  1.25  mi.  and  the  diameter  of  whose  base  is  2.35  miles. 
Find  its  volume  in  cubic  miles. 

23.  How  much  water  will  it  take  to  fill  a  conical  vase  10" 
deep  and  3.225"  across  at  the  top. 

24.  The  conical  tower  of  a  building  is  24.75'  across  at  the 
base  and  36.8'  high.      How  many  cubic  feet  of  space  does  it 
occupy? 

25.  If  a  cord  or  waxed  tape  (bicycle  repair  tape)  be  wrapped 
around  the  curved  outside  surface  of  a  half  croquet  ball,  as  a  top  is 
wound,  until  the  surface  is  covered,  and  then  if  the  same  cord,  or 
tape,  be  wrapped  around  on  the  flat  circular  base  beginning  at  the 
center  until  the  circle  is  covered,  the  length  of  the  former  cord 
will  be  found  to  be  just  twice  the  latter.     The  area  of  the  surface 
of  the  whole  ball  is  then  how  many  times  the  area  of  the  circular 
section  of  the  ball?     What  radius  has  this  circular  section? 

It  is  proved  in  geometry  that  the  area  of  the  surface  of  a  sphere 
equals  4  times  the  area  of  a  circular  section  going  through  the 
center  of  the  sphere. 

26.  The  radius   of  a  ball   is  1|";  what   is  the  area   of  the 
greatest  circular  section  of  the  ball?     What  is  the  area  of  the 
surface  of  the  ball? 

27.  Measure  the  circumference  of   a   baseball  and   find    how 
many  square  inches  of  leather  there  are  in  the  cover  of  the  ball. 

28.  How  many  square  inches  of  paint  will  it  take  to  cover  the 
surface  of  a  globe  of  10"  radius?  of  10"  diameter? 

29.  The  average  diameter  of  the  earth  is  7918  mi. ;  how  many 
square  miles  are  there  in  its  surface? 


320  RATIONAL    GRAMMAR    SCHOOL   ARITHMETIC 

30.  Calling  s  the  surface  and  r  the  radius  of  a  sphere,  write 
an  equation  showing  the  relation  between  s  and  r. 

It  is  seen  on  p.  281  that  the  volume  of  a  triangular  right 
prism  equals  ^  the  volume  of  a  square  prism  whose  base  and  alti- 
tude are  equal  to  the  base  and  the  altitude  of  the  square  prism. 

31.  Find  the  volume  of  a  right 
triangular  prism  whose  altitude  is  18" 
and  whose  base  is  a  triangle  having 
a  base  of  8"  and  an  altitude  of  5" 
inches. 

FIGURE  «T  FIGUBE  208  ff     ft    ^^     pyramid     be     ^ 

fully  modeled  (Fig.  207)  and  filled  with  sand  it  will  be  found  that 
just  3  times  the  volume  of  the  pyramid  is  equal  to  the  volume  of 
the  model  of  a  triangular  prism  (see  Fig.  208)  of  equal  base  and 
equal  altitude.  It  is  proved  in  geometry  that  the  volume  of  any 
pyramid  equals  £  of  the  volume  of  a  prism  having  an  equal  base 
and  an  equal  altitude.  Notice  in  Fig.  208  how  a  triangular  prism 
may  be  completed  on  a  triangular  pyramid  having  the  same  base 
and  the  same  altitude  as  the  prism. 

32.  Calling  V  the  volume,  B  the  area  of  the  base,  and  a  the 
altitude  of  a  triangular  pyramid,  write  an  equation,  showing  the 
way  V  would  be  computed  from  B  and  a. 

33.  Find  the  volume  of  a  pyramid  whose  base  contains  16  sq.  in. 
and  whose  altitude  is  12  inches. 

34.  The  Great  Pyramid  of  Egypt  is  481  ft.  high  and  its  base  is 
a  756'  square.     If  it  were  solid  and  had  smooth  faces,  how  many 
cubic  feet  of  masonry  would  it  contain? 

35.  At  the  close  of  the  nineteenth  century  the  United  States 
had  195,887  mi.  of  railroad.     How  many  times  would  these  rail- 
roads, if  placed  end  to  end,  encircle  the  earth?     (Use  TT  =  3|,  and 
the  radius  of  the  earth  =  3959  miles). 

36.  The  ties  used  for  these  roads  would  contain  wood  enough 
to  make  a  pyramid  of  the  same  shape  1395'  high  with  a  2192' 
square  for  its  base.     How  many  cubic  feet  of  wood  were  used  for 
the  railroad  ties? 

37.  It  has  been  computed  that  the  materials  used  for  the  road 
beds  for  these  railroads  would  make  a  solid  pyramid  2470  ft.  high 


CONSTRUCTIVE    GEOMETRY  321 

and  having  a  3870  ft.  square  for  its  base.     How  many  cubic  feet 
would  this  make? 

If  the  entire  surface  of  a  globe,  or  sphere,  were  divided  up 
into  small  triangles  like  the  one  shown  in  Fig.  209,  and  the  sphere 
were  cut  up  by  planes  cutting  along  the  curved  sides 
of  the  triangles  and  passing  through  the  center,  0, 
the  volume  of  the  sphere  would  be  divided  up  approx- 
imately into  small  triangular  pyramids,  having  their 
vertices  at  the  center.     The  volume  of  each  pyramid 
would  be  the  product  of  its  triangular  base  by  £  of  the      FIGURE  209 
the  radius  of  the  sphere.     The  sum  of  the  a*-  eas  of  all 
triangular  pyramids  would  equal  the  surface  of  the  whole  sphere 
and  the  sum  of  all  the  volumes  of  the  pyramids  would  equal  the  sur- 
face of  the  whole  sphere  multiplied  by  |  of  the  radius  of  the  sphere. 

38.   Calling  Fthe  volume  and  r  the  radius  of  a  sphere,  give 
the  meaning  of  the  formula  : 


39.  Find  the  number  of  cubic  inches  in  a  sphere  of  2"  radius. 

40.  How  many  cubic  inches  in  a  croquet  ball  4"  in  diameter? 
in  a  tennis  ball  1.75"  in  diameter?  in  a  baseball  2.1"  in  diameter? 
in  a  globe  10.15"  in  diameter? 

41.  Calling  the  sun,  the  moon  and  the  planets  all  spheres  with 
diameters  in  miles  as  in  the  following  table,  compute  their  cir- 
cumferences  in  miles,  their  surfaces  in  square  miles  and  their 
volumes  in  cubic  miles: 

Moon  ........  2,160;  Mars  .........  4,230;  Uranus  ......  31,900; 

Mercury  .....  3,  030  ;  Jupiter  ......  86,500  ;  Neptune  .....  34,  800  ; 

Venus  ........  7,700;  Saturn  ......  73,000;  Sun  ........  866,400. 

Earth  ........  7,918; 

§197.  Constructive  Geometry. 

PROBLEM  I.  —  To  find  the  center  of  a  give"n  arc. 

EXPLANATION.  —  Let  AB,  Fig.  210,  be  the  given  arc  whose  center  is  to 
be  found. 

Mark  any  three  points,  as  C,  E,  and  D,  on  the 
arc.  Draw  the  straight  lines  CE  and  ED.  Bisect 
each  of  these  lines  as  in  Problem  VI,  pp.  106  and  107, 
and  prolong  these  bisectors  until  they  intersect  as 
at  O.  O  is  the  required  center. 

DEFINITION.  —  The  lines  CE&ud  ED,  each  of  which 
connects  two  points  of  the  arc,  are  called  chords  of 
FIGURE  216"  the  arc. 


322 


RATIONAL   GRAMMAR    SCHOOL   ARITHMETIC 


To  solve  this  problem  is  it  necessary  actually  to  draw  the 
chords? 

How  could  you  find  the  center  of  a  circle  that  would  go  through 
any  three  points  not  in  a  straight  line?  Mark  3  points  not  all  in 
the  same  straight  line  and  draw  a  circle  through  them. 

PROBLEM  II. — To  bisect  a  given  arc. 

X%  EXPLANATION.— Let  AB,  Fig.  211,  be  the  given  arc. 

»'  \  With  A  as  a  center  and  then  with  B  as  a  center  and 

with  a  radius  greater  than  the  distance  from  A  to  the 
middle  of  the  arc  AB,  draw  the  dotted  arcs  as  indicated. 
Lay  a  ruler  on  the  intersections  of  the  two  arcs  and  draw 
a  short  line  across  the  arc,  as  at  C.  C  is  the  mid-point  of 
the  arc,  and  arc  AC  =  arc  CB. 


y 

FIGURE  211 


PROBLEM  III. — To  circumscribe  a  circle  around  an  equilateral 
triangle. 


EXPLANATION. — Draw  an  equilateral  triangle 
as  in  Problem  VII,  p.  108,  and  bisect  two  of  its 
angles  as  shown  in  Fig.  212.  With  the  intersec- 
tion of  the  bisectors  as  a  center  and  with  a  radius 
equal  to  the  distance  from  this  intersection  to 
any  vertex,  draw  a  circle.  This  circle  is  said  to 
be  circumscribed  around  the  triangle. 

PROBLEM  IV. — To  draw  a  trefoil. 


FIGURE  212 


FIGURE  213 


EXPLANATION. — Draw  an  equilateral  tri- 
angle and  bisect  one  of  its  sides,  as  shown  in 
Fig.  213.  With  the  upper  vertex  as  center  and 
with  a  radius  equal  to  the  distance  from  this 
vertex  to  the  middle  of  the  bisected  side  draw 
an  arc  of  a  circle  around  until  it  touches  the 
sides  of  the  triangle  both  ways.  Draw  arcs 
around  the  other  vertices  in  the  same  way. 

If  desired,  other  circles,  with  slightly 
longer  radii,  may  be  drawn  just  outside  of 
these  until  the  arcs  come  together  but  do  not 
cross. 


PROBLEM  V. — To  construct  a  square  and  to 
draw  a  quatrefoil  (a  four-foil)  upon  it. 

EXPLANATION. — Prolong  BA  through  A  far  enough 
to  draw  a  perpendicular,  as  AC,  to  AB  at  A.  Draw 
this  perpendicular  and  make  AC  =  AB.  With  AB  as  a 
radius  and  (1)  with  C  as  a  center,  then  (2)  with  B  as  a 
center,  draw  two  intersecting  arcs  at  D.  Draw  CD 
and  BD  and  complete  the  drawing  as  shown  in  Fig.  214. 


FIGURE  214 


CONSTRUCTIVE    GEOMETRY 


323 


PROBLEM  VI. — To  draw  the  designs  of  Fig.  215. 


EXPLANATION. — Draw  a  square  like  the 
dotted  squares'of  Fig.  215.  Study  the  two  draw- 
ings, decide  where  the  centers  of  the  arcs  are 
and  complete  the  designs. 


FIGURE  215 


PROBLEM  VII. — To  draw  a  sixfoil. 

EXPLANATION. — Draw  a  circle,  like  the  dotted 
one  in  Fig.  216,  with  a  radius  as  long  as  one  side 
of  the  regular  hexagon  is  to  be.  Draw  the  hexa- 
gon (see  Problem  XI,  p.  110.)  Bisect  a  side  of  the 
hexagon,  and,  using  the  vertices  of  the  hexagon 
as  centers,  complete  the  sixfoil,  as  shown. 

Outside  arcs  may  be  added  if  desired. 


FIGURE  216 


PROBLEM  VIII. — To  draw  a  five-point  star  within  a  circle. 


EXPLANATION.— Draw  a  circle  and  open  the  feet 
of  the  compasses  by  trial  until  5  steps  will  just  reach 
round  the  circumference.  Complete  the  drawing  as 
shown  in  Fig.  217. 

The  strips  need  not  be  interlaced  unless  desired. 
In  this  case  the  inside  lines  need  not  be  drawn. 


FIGURE  217 


PROBLEM  IX. — To  draw  a  regular  pentagon  (five-sided  figure) 
on  a  given  line  as  side. 

EXPLANATION. — Let  the  lower  side  of  the  regular  pentagon,  Fig.  218, 
be  the  given  side.   With  this  side  as  a  radius  draw  the  dotted  half -circum- 
ference  as   shown.     Spread  the  points  of 
the  compasses  by  trial  until  5  steps  of  the 
compasses  will  just  reach  round  this  half- 
circumference.    Draw  a  radius  of  the  dot- 
ted half-circumference  to  the  end  of  the 
third  step. 

Bisect  this  radius  and  also  the  given 
side  with  perpendiculars  and  prolong  the 
perpendiculars  until  they  cross.  Using  the 
crossing  point  as  a  center  and  the  distance 
from  it  to  either  end  of  the  given  side  as  a 
radius,  draw  a  circle.  With  the|  given  side 
as  distance  between  the  feet  of  the  compasses,  and  with  the  intersection  of 
the  two  circumferences  as  a  center,  mark  off  a  point  on  the  full  circum- 
ference, and,  with  this  latter  point  as  a  center,  mark  another  point.  Con- 
nect the  points  as  shown,  thus  completing  the  regular  pentagon. 


FIGURE  218 


324 


RATIONAL    GRAMMAR   SCHOOL   ARITHMETIC 


PROBLEM  X. — To  draw  a  cinquefoil  (a  five-foil). 

EXPLANATION. — Draw  a  regular  pentagon,  bisect  one 
of  its  sides,  and  complete  the  drawing,  as  shown  in 
Fig.  219. 


FIGURE  219 


PROBLEM  XI.  —To  draw  the  design  shown  in  Fig.  220. 


EXPLANATION.  —First  draw  the  three  circles 
touching  each  other  two  and  two  as  in  the  left 
part  of  Fig.  220. 

Noticing  the  thin  center  lines  in  the  design 
draw  the  part  of  the  figure  on  the  right. 


FIGURE  220 


PROBLEM  XII. — To  draw  the  design  of  Fig  221. 

EXPLANATION.— First,  draw  the  large  circle 
of  the  part  of  the  figure  on  the  left.  Then 
draw  two  perpendicular  diameters  and  at  the 
end  Tmake  AT  perpendicular  to  OT  and  as 
long  as  OT.  Draw  OA  and  bisect  the  angle 
OAT  thus  locating  C. 

With  O  as  a  center  and  OC  as  a  radius 
mark  the  centers  of  the  other  three  circles. 
Draw  the  four  small  circles. 

Now  notice  the  fine  center  lines  of  the  part  of  the  figure  on  the  right 
and  draw  it. 


FIGURE  221 


PROBLEM  XIII. — To  draw  the  designs  of  Figs.  222  and  223. 

EXPLANATION.— Study  the  thin  center  lines  and  make  the  sides  of  the 
dotted  square  half  as  long  as  the  sides  of  the  large  square. 

In  both  figures  be  careful  to  have  the  lines  just  touch  but  not  cross. 


FIGURE  222 


FIGURE 


CONSTRUCTIVE    GEOMETRY 


325 


PROBLEM  XIV. — To  draw  the  designs  for  moldings  of  Figs. 
224  and  225. 

EXPLANATION. — Make  enlarged  drawings  of  the  patterns,  being  careful 
to  draw  the  proper  lines  to  locate  the  centers  of  all  the  circular  arcs. 

The  last  pattern  is  a  drawing  of  an  elliptical  molding.  The  figure  will 
explain  how  to  find  the  points  through  which  the  elliptical  curve  is  to  be 
drawn  free-hand. 


FIGURE  224 


FIGURE  225 


1.  Torus.          2.  Scotia.          3.  Ovolo,  or  quarter  round.          4.  Cavetto. 
5.  Cyma  recta.        6.  Cyma  recta.         7.  Echinus,  or  ovolo. 

PROBLEM  XV. — To  construct  a  right  triangle. 

The  symbol  i  means  "perpendicular," 
"perpendicular  to,"  or  "is  perpendicular  to." 

CONSTRUCTION. — Draw  the  line  CD  LAB, 
Fig.  226.  Then  draw  the  line  OF  connecting 
any  point,  as  67,  of  the  line  CD  with  any 
point  of  AB,  as  F. 

DEFINITIONS.— The  longest  side,  that  is, 
the  side  opposite  the  right  angle  of  a  right 
triangle,  is  called  the  hypothenuse. 

What  side  of  the  triangle,  GOF,  is  the  hypothenuse?     What 
angle  is  the  right  angle? 

PROBLEM  XVI. — To  construct  a  right 
triangle  having  a  given  hypothenuse. 

CONSTRUCTION.— Let  the  line  AB,  Fig.  227, 
denote  the  given  hypothenuse.     Bisect  AB  as  at 
O  and  with  O  as  a  center  and  OA  as  a  radius, 
draw  a  semicircle.      Connect  any  point  of  the 
\  /  semi -circumference,   as  E,  D,  or  C,  with  A  and. 

with  B.     Any  such  triangle  is  a  right  triangle 
FIGURE  227  and  the  line  AB  is  its  hypothenuse. 


RATIONAL   GRAMMAR    SCHOOL   ARITHMETIC 


237. 


Point  out  the  right  angle  of  each  of  the  three  triangles  of  Fig. 


FIGURE  228 


PROBLEM  XVII.  —  To  construct  a 
right  triangle,  having  given  the  two  sides 
which  include  the  right  angle. 

CONSTRUCTION.— Let  the  two  given  sides 
be  a  and  b,  Fig.  228. 

Draw  BC  i  DE,  and  make  OA  =  a  and  OB 
=  b.  Connect  A  with  B.  BOA  is  the  re- 
quired triangle. 

How  may  an  isosceles  right  triangle 
be  constructed? 


PROBLEM  XVIII.— To  find  the  relation  of  the  squares  of  the 
sides  of  an  isosceles  right  triangle. 

CONSTRUCTION.— Construct  an  isosceles  right  trian- 
gle and  on  each  of  its  three  sides  draw  a  square.  Draw 
the  dotted  lines  and  cut  the  side  squares  as  shown  in 
£  ig.  ^9.  Fit  the  pieces  over  the  large  square  on  the 
hypothenuse.  If  the  area  of  each  side  square  were  9 
sq.  m.  what  would  be  the  area  of  the  large  square? 

FIGURE  229 

PROBLEM  XIX.— To  find  the  relation  of  the  squares  of  the 
three  sides  of  any  right  triangle. 


FIGURE  230 


CONSTRUCTION. —Construct  any  right  triangle  and  then  construct  a 
square  on  each  of  its  three  sides.  Cut  the  side  squares  as  shown.  Fit 
the  pieces  over  the  square  drawn  on  the  hypothenuse,  as  indicated  by 
the  dotted  lines  in  Fig.  230.  What  single  square  has  an  area  that  equals 
the  sum  of  the  areas  of  the  squares  on  the  two  shorter  sides  of  any  right 
triangle? 


SQUARES   AND   SQUARE    ROOTS  327 

1.  Denoting  the  length  of  either  short  side  of  Fig.  229  by  a, 
what  denotes  the  area  of  the  square  drawn  upon  this  side? 

2.  Denoting  the  length  of  the  hypothenuse  of  Fig   229  by  &, 
what  denotes  the  area  of  the  square  drawn  on  the  hypothenuse? 

3.  Write  an  equation   from   Fig.   229,    showing   the  relation 
between  a?  and  A2. 

4.  Denote  the  lengths  of  the  three  sides  of  any  right  triangle 
(Fig.  230)  by  a,  ft,  and  li  (li  being  the  hypothenuse).     What  will 
denote  the  areas  of  each  of  the  squares  on  the  three  sides? 

5.  Write  an  equation  showing  the  relation  of  the  squares  of 
the  sides  of  any  right  triangle. 

PROBLEMS 

1.  The  sides  of  a    right  triangle  are  3"  and  4";  what  is  the 
length  of  the  hypothenuse? 

SUGGESTION.— 7i2  =  32  -f  42  =  9  +  16  =  25 ;  what  is  the  value  of  h? 

2.  The  hypothenuse  of  a  right  triangle  is  10",  and  one  of  the 
sides  is  6";  what  is  the  other  side? 

SUGGESTION.— 102  =  a2  -f-  6- ;  find  the  value  of  a? 

3.  The  sides  of  a  right  triangle  are  denoted  by  «,  ft,  and  the 
hypothenuse  by  h  •  find  the  unknown  side  in  each  of  the  following 
right  triangles : 

(1)  a  =    9,  ft  =  12;  (4)  a  =  32,  ft  =  18; 

(2)  a  =  12,  li  =  20;  (5)   ft  =  21,  li  =  35; 

(3)  ft  =  15,  U  =  25;  (6)  ft  =  27,  li  =  45. 

Before  the  hypothenuse  of  a  right  triangle,  whose  sides  are  34 
and  26,  can  be  computed,  it  is  necessary  to  know  how  to  find  the 
square  roots  of  given  numbers. 

§198.  Squares  and  Square  Roots. 

DEFINITION. — The  product  obtained  by  using  any  number  twice  as  a 
factor  is  called  the  square  of  that  number  Thus,  36  is  the  square  of  6, 
'because  6,  used  twice  as  a  factor  gives  36  (6x6  =  36).  The  square  of  a 
number  as  6  is  often  written  thus,  62.  What  does  the  small  2  show? 


RATIONAL    GRAMMAR    SCHOOL    ARITHMETIC 

The  following  squares  should  be  committed  to  memory : 

SQUARES  OF  UNITS  SQUARES  OF  TENS  SQUARES  OF  HUNDREDS 

I2  =     1;  102  =    100;  100*  =-10000; 

22=    4;  202  =    400;  2002  =    40000; 

32=     9;  302  =     900;  3002  =     90000; 

42  =  16;  402  =  1600;  4002  =  160000; 

52  =  25;  502  =  2500;  5002  -  250000; 

62  =  36;  602  =  3600;  6002  =  360000; 

72  =  49;  702  =  4900;  7002  =  490000; 

82  =  64;  802=6400;  8003  -  640000; 

92  =  81 ;  902  =  8100;  9002  =  810000. 

1.  Write  all  the  pairs  of  numbers  which,  multiplied  together, 
give   the   product    36;    the   product    16;    the   product    64;    the 
product  49. 

NOTE.— Write  the  pairs  of  factors  of  36  thus:  1  and  36;  2  and  18 ;  3  and 
12;  4  and  9;  6  and  6.  Proceed  similarly  with  the  rest. 

DEFINITION. — The  square  root  of  a  number  is  one  of  the  two  equal 
factors  of  it.  The  sign  of  square  root  is  >/,  called  the  radical  sign.  Thus, 
\/25  means  the  square  root  of  25,  which  is  5. 

2.  Give  the  square  roots  of  the  following  numbers: 

9;  16;  49;  64;  81;  400;  2500;   3600;  160000;  490000;  810000. 

To  find  the  square  root  of  a  number  not  in  the  table  above,  it 
is  necessary  first  to  learn  how  the  square  of  a  number  is  formed 
form  the  number. 

From  the  table  answer  the  following  questions : 

8.  How  many  digits  are  there  in  the  square  of  any  number  of 
units?  of  tens?  of  hundreds? 

4.  Find  the  square  of  each  of  the  folio  wing  decimals:  .1,  .3,  .5, 
.7,  .9,  .01, .03, .04,  .06,  .07, .09, .001, .003,  .005, .006, .007,  .009. 

5.  How  many  decimal  places  are  there  in  the  square  of  any 
number  of  tenths?  of  hundredths?  of  thousandths? 

6.  If,  then,  the  square  of  any  number  of  units  contains  only 
units  and  tens,  what  places  of  any  number  that  is  a  square  must 
contain  the  square  of  the  units  of  its  square  root? 

7.  What  places  must  contain  the  square  of  the  tens?  of  the 
hundreds?  of  the  tenths?  of  the  hundredths?  of  the  thousandths? 


SQUARES   AND   SQUARE    ROOTS 


329 


8.  If,  then,  we  separate  a  number,  whose  square  root  is  desired, 
into  two-digit  groups,  beginning  at  the  decimal  point  and  proceed- 
ing both  toward  the  left  and  toward  the  right,  what  one  of 
these  groups  must  contain  the  square  of  the  units  of  the  square 
root?  the  square  of  the  tens?  of  the  hundreds?  of  the  tenths? 
of  the  hundred ths? 

9    Find  the  square  of  46. 


40 


COMMON  METHOD         MEANING  OF  COMMON  METHOD 
46  =  40+6; 

462  =  (40  +  6)2  =  (40  +  6)  (40  +  6) 
=  40X40  +  40X6+6X40  +  6X6 
=  402  +2  X  (40  X  6)  +  62  =  1600  + 
480+36  =  2116. 

Show  the  meaning  of  the  parts  of 
this  sum  in  Fig.  231. 

Thus  it  is  seen  that  the  square 
_  ire  number  is  the    sum  of  (1)   the 
square  of  the  tens,  (2)  twice  the  product  of  the 
tens  by  the  units,   and  (3)  the  square    of    the 
units. 

This  shows  that  if  any  two-figure  number  be  denoted  by  t  +  u,  where 
t  denotes  the  tens  and  u  the  units,  the  square  is  formed  thus : 

, L 

ml 

t  +  u 


* 

R 

s 

46 

46 

276 

184 

o 

S 

T 

§           2116 
462  =  2116. 

40 

6 

of  a  two-fi| 

xniia.r«   nf   t, 

FIGURE  ssi 


40+6 
40  +  6 


240  +  36 

1600  +  240 

1600  +  480  +  36 


p  +  2tu  +u*  =  (t  +  uf      <- 


R 

sl: 

S 

T 

t 

II 

FIGURE  232 

Show  the  meaning  of    the    equation  by 
Fig.  233. 


10.  Find  the  square  root  of  2116. 

CONVENIENT  FORM 
46  =  required  root ; 
2116    denoted  by  t2  +  2tu  +  u2 ; 
1600  =  greatest  square  (of  the  table)  in  2116; 
2t  =  80     I  516    contains  2tu  +  u2,  where  t  =  40 ; 
2t  +  u  =  86     |  516    denoted  by  2tu  +  w2  ,  where  u  =  6 ; 

Check:  46  X  46  =  2116. 

SHORTENED  FORM 
21 '16  I  46  =  required  root 

80     16 

Check:  46  X  46  =  2116. 


516 
,516 


330 


RATIONAL    GRAMMAR   SCHOOL    ARITHMETIC 


11    Find  the  square  roots  of  the  following  numbers : 

(1)484;     (3)1156;     (5)1296;     (7)7569;     (9)9604; 
(2)    729;     (4)1225;     (6)5625;     (8)  9409;   (10)  110224. 

12.  Find  the  square  root  of  2079.36. 


20'79.36'|45.6 
16'00      


HO 


90. 


4'79. 
4'25. 


54.36 
54.36 


v/2079.36  =  45.6 
Check:  45.6X45.6  =  2079.36. 


CONVENIEI^T  FORM 

EXPLANATION. — Separate  the  number 
into  two-digit  groups.  Beginning  on  the 
extreme  left,  subtract  the  greatest  square 
of  tens  in  20  hundreds,  viz.,  16  hundreds 
(=402),  and  write  4  tens  on  the  right  as 
the  first  root  digit.  Double  the  4  tens  and 
use  the  result  80  as  a  trial  divisor.  80  is 
contained  in  the  remainder,  479,  5  times. 
Write  5  as  the  2d  root  digit  and  also  add  it 
to  the  80,  giving  85  as  the  complete  divisor. 
Double  the  part  of  the  root  found,  45, 
giving  90  for  the  next  trial  divisor  and 
complete  the  steps  as  before. 


13.   Find  the  square  root  of  1578  to  3  decimal  places. 


CONVENIENT  FORM 


60 


78.0 
.7 


78.7 
79.40 

.02 
79.42 
79.440 

.004 

79.444 


15'78.00'00'00'139.724-f- 
9 

1678. 
1621. 

57.00 
55.09 


1.9100 
1.5884 

.321600 

.317776 

.003824  remainder. 


EXPLANATION. — Annex  zeros  and 
proceed  as  before. 

Check:  39.724  X  39.724=1577.996176. 
rem.  =        .003824. 


1578. 


In  actual  practice  the  decimal  point  is  needed  only  in  the  root. 
14.  Find  the  square  roots  of  the  following  numbers : 


(1)  1900.96; 

(2)  4719.69; 

(3)  5055.21; 


(4)  61.1524; 

(5)  75.8641; 

(6)  79.9236; 


(7)  .5476; 

(8)  .458329; 

(9)  1.216609. 


SQUARES    AND    SQUARE    ROOTS  331 

15.  Find  to  3  decimal  places  the  square  roots  of  the  following 
numbers  : 

(1)  5;          (4)     .85;          (7)   1683;          (10)     1.85; 

(2)  7;         (5)     .25;         (8)   6875;         (11)  26.79; 

(3)  15;         (6)     125;         (9)  7328;         (12)  64.893. 

16.  Find  by  multiplication  the  values  of  the  following  expres- 
sions: 

(1)  (i)2;       (3)  (I)2;       (5)  (T\)2;      (7)  (|-)2; 

(2)  (f)2;        (4)  (if)2;      (G)  (B)2;       (8)  (f)2. 

17.  Make  a  rule  for  finding  the  square  root  of   a   common 
fraction. 

18.  Find,  without  reducing  the  common  fractions  to  decimals, 
the  values  of  the  following  expressions  and  prove  them  by  multi- 
plication : 

(i)  ,/i;  (*) 

(2)  y/f;  (5) 

(3)  ,/;  (6) 


19.  The  sides  of  a  right  triangle  are  respectively  34"  and  26" 
long,  how  long  is  the  hypothenuse? 

20.  The  center  pole  of  a  circus  tent  is  35'  high,  and  a  guy 
rope  is  stretched  from  the  top  of  the  pole  to  a  stake  56'  from  the 
bottom.     How  long  is  the  rope,  supposing  the  ground  level  and 
the  rope  straight,  allowing  4'  for  tying? 

21.  50'  of  the  top  of  a  tree  standing  on  level  ground  is  broken 
by  the  wind  and  remains  fastened  to  the  stump.      If  the  top 
strikes  the  ground  30'  from  the  stump,  how  high  was  the  tree? 

22.  A  horse  is  staked  out  by  a  rope  40'  long  to  the  top  of  a 
stake  15"  high.     Over  what  area  can  the  horse  graze? 

23.  The  vertical  mast  of  a  hoisting  derrick  35'  high  is  held 
in  position  by  four  guy  ropes  staked  to  the  ground  at  the  four 
corner  points  of  a  square.     The  stakes  are  75'  from  the  bottom 
of  the  mast.     How  much  will  the  rope  for  the  4  guys  cost  at  2.5^ 
a  foot,  20'  being  allowed  for  knots? 

24.  What  is  the  area  of  the  square  whose  corners  are  at  the 
stakes? 


332  RATIONAL   GRAMMAR   SCHOOL   ARITHMETIC 

§199.  Square  Root  of  Numbers  and  Products  Geometrically. 

1.  Find  the  square  root  of  2  geometrically. 

CONSTRUCTION. — To  any  convenient  scale  draw  a  line  2-f-  1  (=3)  units 
long.  Bisect  the  line  and  draw  a  semicircle  upon  it 
as  a  diameter  (Fig.  283).  At  the  point  of  division 
between  the  second  and  third  unit,  draw  a  perpendicu- 
lar to  the  diameter.  The  part  of  this  perpendicular 
between  the  diameter  and  the  circumference  is 


FIGURE  233         v/a  (=  1.414  _j_)  units  long  to  the  scale  used. 

2.  Draw  the  square  root  of  7. 

SUGGESTION.— Proceed  as  before,  taking  a  diameter  7  +  1  (=  8)  units 
long  and  drawing  the  perpendicular  at  the  point  of  division  at  the  end  of 
the  7th  unit. 

3.  Draw  the  square  root  of  each  of  the  following  numbers : 

5;     11;     13;     17;     19. 

4.  Find  the  square  root  of  4  x  3,  or  12,  geometrically. 

CONSTRUCTION. — Draw,  to  a  convenient  scale,  a 
line  AB  (Fig.  234)  7  units  long.  Draw  a  semicircle 
on  AB  as  a  diameter,  and  at  C,  the  end  of  AC  (  =  4), 
draw  a  perpendicular  CD.  The  part  of  the  perpen- 
dicular between  the  circumference  and  the  diameter 
represents  the  square  root  to  the  scale  used.  FIGURE  234 

5.  Find  the  square  roots  of  these  products  geometrically: 
(1)  3  x  2,  or    6;        (4)  2  x  4;        (7)   28;       (10)   18; 
(2)3x5,  or  15;       (5)7x3;       (8)32;       (11)26; 
(3)  4  x  5,  or  20;       (6)   6  x  5;       (9)   40;       (12)  27. 

§200.  Cubes  and  Cube  Roots. 

1.  What  is  the  volume  of  a  cube  whose  edge  is  5"?  6"?  7"? 
11"?  18"?  21"?  24"? 

DEFINITION.— The  cube  of  a  number  is  the  product  obtained  by  using 
the  number  3  times  as  a  factor.  Thus  the  cube  of  4  is  4  X  4  X  4  =  43  =  64- 

Table  of  cubes  of  numbers  to  be  learned : 

CUBES  OF  UNITS  CUBES  or  TENS 

I3  =      1;  10s  =      1000; 

23=      8;  203=      8000; 

33=    27;  303=    27000; 

43=    64;  403=    64000; 

5s  =  125;  503  =  125000; 

63  =  216;  603  =  216000; 

7s  =  343;  W  =  343000; 

83  =  512;  803  =  512000; 

93  =  729;  903  =  729000. 


CUBES   AND    CUBE    ROOTS 


333 


DEFINITION. — The  cube  root  of  a  number  is  one  of  its  3  equal  factors. 
Cube  root  is  indicated  by  the  sign  i/7.  Thus,  t/7729  means  one  of  the 
three  equal  factors  of  729,  which  is  9.  The  sign  i/77s  called  a  radical  sign. 

2.  Between  what  two  whole  numbers  are  the  following  cube 
roots : 

iXIC?   i/x35?   iX?8?   i/450?   ^075?    £/895?  v7 58000?  i/ 480000? 

3.  Find  by  multiplication  the  values  of  the  following  cubes : 

4.  Make  a  rule  for  finding  the  cube  of  any  common  fraction, 
o.  Prove  the  following  relations  by  multiplication: 

K/2107  =  13 ;    5/4096 


16; 


PHtt-«5 

G.  Make  a  rule  for  finding  the   cube   root   of   any  common 
fraction. 

7.  Find  the  cube  roots  of  the  following: 

-1  •     27..     12.6  .       7_2_9_  •    _3_4_3_  •          UL9__ 
¥  J     64J    ¥41?  >     8000J     1000J    TS  5  0  0  0  * 

§201.  Triangles  Haying  the  Same  Shape  (Similar  Triangles) . 

1.  Write  the  numerical  values  of  the  follow- 
ing ratios  from  Fig.  235 : 

^  ~T6';      ^          '     '** 


FIGURE  235 

2.  In   Fig.  236  the  triangles   have   the  same  shape.     Write 
the  values  of  the  ratios : 


3.  In  Fig.  237  the  triangles 
have  the  same  shape,  and  A  C  = 
7  x  ac.  Find  the  sides  of  the 
triangle  ABC,  if  ac=  tV';  «*  = 
i",  and  Jc-iJ"- 


FIGURE  236 


FIGURE  237 


334 


RATIONAL   GRAMMAR   SCHOOL   ARITHMETIC 


4.  Supposing  that  ac  (  =  }  AC)  represents  7'  (Fig.  237),  ab, 
10'  and  be,  11',  what  lengths  do  AC,  AB,  and  BC  represent? 

5.  In  Figs.  238  and  239,  AB  =  4«J;  if  A B represents  1  mi.,  BC, 
3  mi.,  and  AC,  3|  mi.,  what  distances  do  ab,  be,  and  ac  represent? 


b 

b 

i^ 

^-—^. 

c 

y- 

,--•: 

*-=-^ 

AL 

A/ 

•^-^ 

— 

B 

p 

FIGURE  238 


FIGURE  239 


6.  Draw  a  triangle  having  sides  of  1",  }",  and  J",  and  another 
having  sides  of  3",  2J",  and  1|".     Call  the  angles  opposite  the 
sides  1",  |",   and  |",  a,   #,  and  c,  respectively,  and  the  angles 
opposite  the  sides  3",  2J",  and  1J",  ^4,  .5,  and  (7,  respectively. 
Do  these  triangles  have  the  same  shape?     Carefully  cut  out  the 
triangles  and  place  angle  a  over  angle  A  ;  then  angle  b  over  angle 
B-,  and,  last,  angle  c  over  C?   What  do  you  find  to  be  true  in  each 
case? 

DEFINITION. — In  triangles  having  the  same  shape,  angles  lying  oppo- 
site proportional  sides  in  different  triangles  are  called  corresponding 
angles. 

7.  Read  the  pairs  of  corresponding  angles  in  Figs.  235,  236, 
237,  and  239. 

8.  In  triangles  having  the  same  shape,  how  do  corresponding 
angles  compare  in  size  (see  Problem  6)  ? 

9.  In  Fig.  235  find  the  ratio  of  the  side  lying  opposite  the  angle 
B  in  triangle  ABC  to  the  side  lying  opposite  the  corresponding 
angle  in  triangle  abc.     Find  a  similar  ratio  between  any  other  pair 
of  such  sides.     How  do  the  ratios  compare? 

10.  Answer  similar  questions  for  the  triangles  of  Figs.  236, 
237,  and  239. 

DEFINITION. — In  triangles  haying  the  same  shape,  sides,  as  AB  and 
ab  (Fig.  239)  or  BC  and  be,  that  lie  opposite  the  equal  angles  are  called 
corresponding  sides. 

11.  In  triangles  having  the  same  shape  state  a  law  connecting 
the  corresponding  sides. 


USES    OF    SIMILAR    TRIANGLES 


335 


FIGURE  240 


B  12.  Triangles  ABC  and  ade  (Fig.  240)  have  the 

same  shape.  How  do  their  corresponding  angles 
compare?  What  relation  holds  for  their  cor- 
responding sides,  i.e.,  how  do  these  ratios  compare: 
AB-.adl  BC:de?  ACiae?  With  a  protractor 
measure  each  pair  of  corresponding  angles.  How 
do  they  compare  in  size? 

13.  What  is  the  ratio  of  the  areas  of  triangles  AB 0  and  ade 
(Fig.  240)?     How  may  this  ratio  be  found  from  the  ratio  of  a  pair 
of  corresponding  sides,  as  AB  and  ad? 

14.  Draw  a  pair  of  triangles,  one  having  sides  of  1",  1-J-",  and 
If",  and  the  other  having  sides  of  4",  6",  and  7".     Give  the  ratio 
of  each  pair  of   corresponding  sides;    the  ratio  of  the   areas  of 
the  triangles?     How  may  the  ratio  of  the  areas  be  computed  from 
the  ratio  of  a  pair  of  corresponding  sides? 

15.  State  a  law  connecting  the  ratio  of  the  areas  of  triangles 
having  the  same  shape  with  the  ratio  of  any  pair  of  corresponding 
sides. 


§202.  Uses  of  Similar  Triangles. 

In  Fig.  241  MNQR  represents  an 
B  18"x20"  board,  provided  with  a  mov- 
ing arm,  OE,  and  a  protractor. 
The  arm  is  made  of  light  wood.  A 
slot  E8  is  cut  away  and  a  thread  is 
stretched  on  its  under  side  and  fastened 
at  the  ends.  The  ends  of  the  thread 
are  sunk  into  the  wood  so  as  to  permit 
the  thread  to  move  smoothly  over  the 
protractor  as  the  arm  CO  is  turned 
round  0.  A  common  pin  stuck  ver- 
tically at  C  and  at  0  completes  the 
apparatus.  The  pin  at  0  should  be  driven  through  far  enough 
to  fasten  the  arm  to  the  board  at  0  as  a  pivot.  The  protractor 
is  held  by  thumb  tacks  at  T,  T.  The  board  maybe  placed  on  a 
window  sill,  or  on  a  post  in  a  fixed  position,  while  measure- 
ments are  being  made.  It  is  desired  to  measure  the  distance  from 
A  to  B,  A  and  B  being  on  opposite  sides  of  a  large  building. 

Use  the  proportion  form  of  statement  in  all  problems,  calling 
the  length  of  the  unknown  line  x.     Then  find  the  value  of  x. 


Q 

FIGURE  211 


336  RATIONAL    GRAMMAR    SCHOOL    ARITHMETIC 

Solve  some  problems  like  this  from  your  own  measures.     Build- 
ings, steeples,  hills,  trees,  furnish  good  problems. 

1.  The  board  was  placed  on  a  window  sill  and  the  arm  set  in 
such  position,  FO,  that  when  the  eye  sighted  along  the  pins,  0 
and  Z>,  these  pins  were  in  line  with  B.     The  mark  on  the  pro- 
tractor at  G  was  52°.     The  board  being  held  firmly  in  place,  the 
arm  was  swung  around  0  until  the  pins  at  0  and  at  C  were  sighted 
into  line  with  A  and  the  reading  on  the  protractor  at  H  was  114°. 
How  many  degrees  were  there  in  the  angle  EOF? 

2.  The  distances  from  ^1  and  from  B  to  the  window  at  0  were 
measured  and  found  to  be  484'  and  520'.     A  triangle  O'ef  (shown 
on  the  left)  was  drawn  having  angle  e(9'/'=G20,  and  O'e  =  4.84", 
O'f  -  5.20".     The  side  ef  was  then  drawn,  measured,  and  found 
to  be  5.18".     How  long  is  the  line  ABt 

3.  Fig.  242  shows  a  crude  apparatus  for  finding  distances  that 
cannot   be  measured  directly.     It  consists  of  a  board  about  25" 
long  and  6"  wide  with  end  pieces  about  6"  high.     The  wood  is  cut 


ElGURE  242 

away  from  the  end  pieces  so  that  the  eye  may  sight  through 
between  the  straight  edges  of  two  visiting  cards  at  E,  past  two 
parallel  threads  at  T,  set  carefully  £"  apart  and  25"  in  front  of  E. 

If  8  is  in  line  with  the  upper  thread  of  the  instrument  and  D 
with  the  lower  thread,  how  far  is  it  from  the  stake  at  SD  to  the 
instrument,  if  8D  =  8'6"? 

SUGGESTION.— * :  25  =  8| :  x.   By  the  Principle  on  p.  197,  \x  =8J  X  25,  or 

^  =  2121.     Find  x, 

o 

4.  A  pupil  sighted  through  E  at  a  stake  held  by  another  pupil 
at  8D.  When  the  pupil  at  E  sighted  over  the  top  thread,  the 


USES    OF    SIMILAR   TRIANGLES 


337 


pupil  had  to  put  his  hand  at  #  to  line  in  with  the  slit  at  E  and 
upper  thread  at  T.  The  pupil  at  E  then  beckoned  him  to  slide 
his  hand  down  the  pole  until  it  lined  in  with  the  lower  thread 
at  T.  The  distance  SD  was  7.52'.  How  far  was  it  from  the 
instrument  to  the  flagpole  SD? 

5.  The  distance  from  E  to  the  bottom  of  the  building,  Fig.  243 


FIGURE  243 

is  288',  the  distance  EB  =  8'  and  AB  =  18";   how  high  is  the 
building? 

6.  In  Fig.  244  it  is  desired  to  find  the  distance  across  the  lake 
from  'O  to  D.  A  surveyor  stuck  a  flagpole  at  a  place  from  which 
he  could  see  both  C  and  D.  He  measured  the  distances  from  D 
and  from  C  to  the  pole  he  is  holding,  and  found  them  to  be  4563' 


FIGURE  244 


and  5481'  respectively.  He  then  set  a  pole  at  B,  T^  of  the  dis- 
tance from  the  first  pole  to  Z>,  and  another  pole  at  A,  T^  of  the 
distance  from  the  first  pole  to  C.  The  distance  from  A  to  B  was 
measured  and  found  to  be  84.64'.  How  far  is  it  from  C  to  Z>? 


338 


RATIONAL    GRAMMAR    SCHOOL   ARITHMETIC 


7.  It  is  desired  to  make  a  scale  drawing  of  the  tract  of  ground 
ABODEF  (Fig.  245).  The  parts  of  the  apparatus  to  be  used,  as 
shown  at  the  right,  are  a  square  board  about  16"xl6",  with  a 


block  containing  a  1"  hole  to  fit  over  the  top  pin  of  a  sharp 
stake  to  be  stuck  in  the  ground  to  support  the  board.  A  small 
level  (which  may  be  a  phial  of  water)  and  a  foot  rule  with  a  vertical 
pin  at  each  end  for  sights  to  be  used  for  sighting  complete  the 
apparatus. 

The  board  is  set  up  in  the  field  as  shown,  a  sheet  of  paper 
is  pinned  on  it  with  thumb  tacks,  and  the  foot  rule  is  placed  upon 
the  paper.  A  third  pin  is  stuck  near  the, center  of  the  board  at  a 
point  o  (not  shown  in  Fig.  245).  Holding  the  edge  of  the  foot 
rule  against  this  center  pin,  sighting  along  the  pins  toward  a  pole 
at  A  the  observer  turns  the  front  of  the  foot  rule  until  the  two 
pins  of  the  ruler  are  in  line  with  A.  He  holds  the  ruler  and 
draws  a  line  along  its  edge  on  the  paper. 

He  now  holds  the  edge  of  the  ruler  against  the  center  pin, 
and  carefully  sights  the  two  ruler  pins  into  line  with  B,  and,  hold- 
ing the  ruler  in  place,  draws  a  line  on  the  paper  toward  B.  He 
proceeds  in  the  same  way  with  each  of  the  points  (7,  />,  E,  and 
F,  being  careful  not  to  turn  the  board  around  on  the  stake  pin. 

8.  The  lines  from  the  stake  supporting  the  board  to  A,  to  B, 
to  (7,  and  so  on  to  F  were  measured  and  found  as  follows :  to  A , 
460';  to  B,  452';  to  (7,  378';  to  Z>,  527';  to  E,  535'  and  to  F,  832'. 
Using  a  scale  of  1":  100',  the  distance  to  A  (460')  was  laid 


USES   OF   SIMILAR   TRIAXGLES  339 

off  from  the  center  pin  on  the  line  drawn  toward  A  giving  a  (on 
the  board),  the  distance  to  B  (452')  was  laid  off  from  the  center 
pin  on  the  line  drawn  toward  B  giving  I  (on  the  board),  and  so 
on  around  to  F.  Calling  the  center  pin  0,  how  long  is  oa?  ob? 
oc?  od?  oe?  of? 

9.  With  a  ruler  the  points  «,  #,  c,  d,  e  and  /  were  connected 
as  shown  in  the  figure.  The  lines  were  ab  =  2.8"  ;  be  =  4.5"  ;  cd  = 
5.12";  de  =  2.95";  ef=  6.25";  /«  =  4.48".  How  long  are  the  lines 

,  EC,  CD,  DE,  EF,  and  FA? 


The  board  may  be  supported  by  a  light  camera  tripod  or  by  a 
home-made  tripod  and  the  board  may  be  held  to  the  flat  top  of  the 
tripod  by  a  thumb  nut.  (See  Fig  246.) 

10.  The  distances  from  the  apparatus  to  the  corners  of  the  field 
(Fig.  246)  were:  to  A,  678';  to  B,  612';  to  <7,  683';  to  D,  738'; 


to  E,  698';  to  F,  625';  to  £,  679'.  Using  a  scale  of  1":200', 
how  long  should  the  distances  be  made  from  the  center  pin  to  a? 
to  M  to  c?  to  d?  to  e?  to/?  to  g? 

11.  With  a  ruler  a,  I,  c,  d,  e,  /,  #,  and  a  were  then  connected 
and  lines  measured.  The  measures  were :  0^  =  3.81";  £c  =  4.12"; 
c^  =  2.86";  ^e  =  2.75";  e/=5.86";  /#  =  6.18"  and  ^  =  5.98". 
How  long  are  the  lines  AB?  BC?  CD?  DE?  EF?  FG?  OA? 


RATIONAL    GRAMMAR   SCHOOL   ARITHMETIC 

12.  Having  made  the  scale  drawings,  perpendiculars  may  be 
drawn  from  the  center  pin  to  each  of  the  sides  ab,  b'i  and  so  on. 
From  the  measured  lengths  of  these  perpendiculars  and  the  bases 
AB,  BC,  and  so  on  of  the  triangular  parts  of  the  figures,  the  areas 
of  the  triangles  aob,  boc,  and  so  on  (calling  o  the  point  where  the 
center  pin  stands)  may  be  computed.     The  sum  of  these  areas 
gives  the  total  area  of  the  figure  abcdefga. 

13.  Suppose  the  area  of  triangle  aob  were  4.94  sq.  in.  what 
would  be  the  area  of  A  OB  (0  being  the  point  on  the  ground  just 
under  o),  if  the  scale  were  1":200'?     (See  Problems   13  and  15, 
§201). 

14.  If  the  areas  of  the  triangles  aob,   boc,  cod,  doe,  eof,  and 
foa  (Fig.  245  prob.  7)  in  square  inches  were:  6.073,  7.740,  9.172, 
7.490,  16.725  and  7.563,  respectively,  and  the  scale  were  1":  100'; 
what  were  the  areas  of  the  triangles  AOB,  BOC,  COD,  DOE, 
EOF,  and  FOA? 


APPLICATIONS   OF   PERCENTAGE 
§203.  Insurance. 

1.  What  will  it  cost  to  insure  $680  worth  of  household  furni- 
ture, at  the  rate  of  If  %? 

DEFINITION. — The  amount  paid  for  insurance  is  called  the  premium. 

2.  An  art  gallery,  valued  at  $500,000,  is  insured  at  the  rate  of 
li%-     What  is  the  premium? 

DEFINITIONS. — The  written  agreement  between  an  insurance  company 
and  the  insured  is  called  a  policy. 

The  amount  for  which  the  property  is  insured  is  the  face  of  the  policy. 

3.  A  vessel,  valued  at  $16,000,  was  insured  for  £  of  its  value. 
Find  the  face  of  the  policy. 

4.  A  growing  crop  was  insured  at  5%.     The  premium  was 
$140.     What  was  the  face  of  the  policy? 

5.  If  it  costs  $420  to  insure  a  house  for  f  of  its  value,  at  3J-%, 
what  is  the  house  worth? 


APPLICATIONS    OF    PERCENTAGE  341 

6.  A  stock  of  hardware  was  insured  for  $7000,  insurance  cost- 
ing $110.75.     What  was  the  rate  of  insurance;' 

7.  If  a  shipment  of  grain  is  worth  $840,  and  the  premium 
amounts  to  $17.60,  what  is  the  rate  of  insurance? 

8.  A  machinist  insured  his  tools,  valued  at  $300,  for  J  of  their 
value,  paying  $8.19.     What  rate  did  he  pay? 

9.  A  farmer  paid  a  premium  of  $10.50  for  insuring  his  stock 
at  l-J-%.     For  what  amount  was  the  stock  insured? 

10.  A  man  paid  $67.50  for  the  insurance  of  a  steam  launch  at 
1-J%.     For  what  amount  was  the  launch  insured? 

11.  What  is  the  rate  paid  for  insuring  a  bridge,  valued  at 
$15,000,  for  |  of  its  value,  the  premium  being  $240? 

12.  A  library,  worth  $28,000,  was  insured  for  f  of  its  value, 
the  premium  being  $720.     What  was  the  rate  of  insurance? 

13.  The  contents  of  a  grain  elevator  were  insured  at  a  rate 
of -f%.     What  was  the  amount  of  the  policy,  the  premium  paid 
being  $272.40? 

14.  The  premium  paid  for  insuring  a  quantity  of  lumber,  for 
two-thirds  of  its  entire  value,  at  3% ,  was  $36.    What  was  the  value 
of  the  lumber? 

15.  A  piece  of   property,  valued     at    $27,200,  was    insured 
for  g  of  its  value.     The  premium  was  $212.50.     What  was  the 
rate? 

16.  A  tank  of  oil,  holding  2592  gal.,  worth  I8<p  per  gallon, 
was  insured  at  4%.     Find  the  premium. 

17.  A  man  insured  a  cargo  worth  $2400  at  3%%.     f  of  the 
cargo  was  lost  at  sea.     What  amount  of  insurance  should  the 
man  obtain?     What  premium  had  he  paid? 

18.  What  is  the  rate  of  insurance  paid  for  insuring  26,000  bu. 
of  grain  worth  77^,  for  f  of  its  value,  if  the  premium  is  $500.50? 

19.  For  what  sum  must  a  policy  be  issued  to  insure  a  factory, 
valued  at  $21,000,  at  If  %  ;  a  house,  worth  $28,000,  at  i%  ;  and  a 
barn,  valued  at  $5600,  at  If  %? 

20.  A  steamboat  had  a  cargo,  valued  at  $2000,  which  was 
destroyed  by  fire.     The  insurance  was  $1500.     What  per  cent  of 
the  value  of  the  cargo  was  covered  by  insurance?     What  was  the 
premium,  at 


342  RATIONAL    GRAMMAR    SCHOOL    ARITHMETIC 

§204.  Taxes. 

DEFINITION. — A  tax  is  a  sum  of  money  levied  by  the  proper  officers 
to  defray  the  expenses  of  national,  state,  county,  and  city  governments, 
and  for  public  schools  and  public  improvements. 

1.  A  town  wishes  to  raise  $23,500  to  build  a  public  hall.     If 
the  collector's  commission  is  2£%,  what  is  the  total  amount  to  be 
collected? 

2.  If  the  taxable  property  of  a  town  is  valued  at  $923,846 
and  the  rate  of  taxation  is  3f%,  what  is  the  whole  amount  of 
the  tax? 

3.  The  taxable  property  of  a  town  is  $869,472,  and  the  rate 
is  8£  mills  on  the  dollar.     What  is  the  amount  of  the  tax? 

4.  What  amount  of  money  must  be  collected  to  raise  a  net  tax 
of  $643,000,  allowing  6%  for  collection? 

5.  The  tax  in  a  city  is  $49,682;  the  rate  is  15  mills  on  the 
dollar.     What  is  the  assessed  valuation? 

DEFINITION — Assessed  valuation  means  the  estimated  value  of  the 
property  that  is  assessed. 

6.  What  is  the  rate  of  taxation  when  property  assessed  at  $8940 
pays  a  tax  of  $160.92? 

7.  The  net  amount  collected  as  a  tax  was  $2896.42.     The 
collector's  commission  was  3|%.     What  was  the  whole  amount 
collected? 

8.  The  assessed  valuation  of  the  taxable  property  of  a  town 
was  $1,218,694.     The  tax  to  be  raised  was  $18,280.41.      How 
many  mills  on  the  dollar  was  the  rate  of  taxation? 

9.  A  tax  collector  received  $175.18  as  his  2%  commission  on  a 
certain  sum  collected.     The  assessed   valuation    of   the   taxable 
property  was  $922,000.     What  was  the  tax  of  a  man,  whose  prop- 
erty was  valued  at  $18,000? 

10.  The  real  property  (houses,  lands,  etc.)  of  a  certain  town 
is  valued    at  $4,560,800,  and  the    personal    property  (movable 
property)  at  $945,900.     If  $12,000  is  to  be  raised   by   taxation, 
how    much    must    a    man    pay,    whose    property   is    valued    at 
$9,840? 

11.  A  net  tax  of  $8965  is  to  be  raised  in  a  certain  city.     The 


APPLICATIONS   OF,  PERCENTAGE  343 

assessed  valuation  of  the  taxable  property  is  $5,689,243.  The 
collector's  commission  is  1-|%.  What  will  be  the  tax  of  a  man, 
whose  property  is  valued  at  $56,000? 

12.  Property  on  a  city  street  is  assessed  2%  on  its  valuation. 
How  much  more  will  a  man  pay  who  owns  100  front  feet,  valued 
at  $125    per    foot,    than    one    who    owns    property    valued    at 
$7500? 

13.  If  the  assessed  valuation  of  the  taxable  property  in  a  cer- 
tain city  is  $1,069,210,  and  the  whole  amount  of  tax  collected    is 
$13,899.73,  what  is  the  rate  of  taxation  and  the  net  amount  after 
deducting  the  collector's  commission  of  1J%? 

14.  The  taxable  property  in  a  town  is  $872,990.     The  rate  of 
taxation  is  .015.     What  is  the  net  amount  collected  after  deduct- 
ing the  collector's  commission  of  2J%? 

15.  The  assessed  valuation  of  the  taxable  property  in  a  certain 
city  is  $4,968,390.      The  tax  collected  amounts  to  $94,399.41. 
How  many  mills  on  the  dollar  was  the  rate  of  taxation? 

16.  A  certain  town  wishes  to  raise  by  taxes  $20,500  to  build 
a  schoolhouse.     What  tax  must  be  levied  to  cover  this  and  the 
cost  of  collection  at  4%? 

§205.  Trade  Discount. 

1.  A  hardware  dealer  bought  the  goods  named  in  the  follow- 
ing bill : 

(1)  2  doz.  braces,  @  45^  each; 

(2)  50  Ib.  f"  bolts,  @  3i#; 

(3)  12  bales  barb  wire,  @  $2.15; 

(4)  25  kegs  nails,  @  $2.50; 

(5)  8  cooking  stoves,  @  $28.75; 

(6)  12  baseburners,  @  $32.50; 

(7)  150  Ib.  screws,  @  $4.75  per  cwt. 

The  bill  was  subject  to  discounts  of  10%,  7%,  and  also  5%  30 
da.  or  6%  10  da.  What  was  the  total  cost  if  the  bill  was  paid  in 
30  da.?  in  10  days? 

2.  Find  the  amount  needed  to  settle  the  following  bill,  sub- 


344  RATIONAL    GRAMMAR    SCHOOL   ARITHMETIC 

ject  to  the  successive  discounts,  10%,  8%,  and  5%  off  for  30  da. 
or  6%  for  cash,  the  bill  being  paid  in  cash: 

(1)  2  chests  carpenter's  tools,  @  $48.50; 

(2)  2  doz.  augurs,  @  25^  each; 

(3)  2  doz.  carpenter's  rules,  @  $1.80; 

(4)  480  Ib.  hinges,  @  $4.75  per  C; 

(5)  3  doz.  tin  buckets,  @  22^  each; 

(6)  4  doz.  gardening  rakes,  @  28^  each; 

(7)  14  scoop  shovels,  @  85^-; 

(8)  C  wash  boilers,  @  78^; 

(9)  12  washtubs,  @  52^; 

(10)     10  doz.  clotheslines,  @  $1.10. 

3.  What  amount  will  be  needed  to  settle  the  bill  of  problem  2 
in  30  days? 

4.  What  amount  will  settle  the  following  bill  of  house  furnish- 
ings, the  bill  being  subject  to  the  discounts,  20%,  10%,  8%,  and 
5%  30  da.,  6%  cash,  the  bill  being  paid  in  30  da.? 

(1)  8  dining  tables,  @  $30; 

(2)  12  sets  dining-room  chairs,  @  $18.50  a  set; 

(3)  10  bookcases,  @  $12.50; 

(4)  8  rocking  chairs,  @  $12.75; 

(5)  15  center  tables,  @  $15; 

(6)  12  Wilton  rugs,  @  $25; 

(7)  15  Axminster  rugs,  @  $23; 

(8)  250  yd.  ingrain  carpet,  @  45^. 

5.  What  amount  paid  in  cash  will  be  needed  to  settle  the  bill? 

6.  What  amount  in  cash  will   settle   this  bill  of   plumber's 
supplies : 

(1)  40'  lead  pipe,  @  15#,  discounts  20%,  10%,  7%,  2% 

cash; 

(2)  100'  iron  tubing,  @   3^,  discounts  20%,  10%,  7%, 

2%  cash; 

(3)  8  bathtubs,  @  $20,  discounts  20%,  7%,  2%  cash; 

(4)  8  lavatory  outfits,  @   $10,  discounts  20%,  7%,  2% 

cash, 

(5)  6  water  meters,  @  $5,  discounts  20%,  12%,  2%  cash; 

(6)  4  pumps,  @  $25,  discounts  20%,  7%,  2%  cash; 

(7)  5  pump  valves,  @  $2.85,  discounts  20%,  7%,  2% 

cash? 


APPLICATIONS    OF    PERCENTAGE 


345 


§206.  Stocks  and  Bonds. 

A  Stock  is  a  written  agreement  (called  also  a  stock  certificate), 
made  by  a  company  to  pay  the  holder  a  certain  part  of  the  earn- 
ings of  the  company.  The  sum  paid  is  reckoned  at  a  certain 
number  of  dollars  per  share,  or  at  a  certain  rate  per  cent  of  the 
face  value  of  a  share.  A  snare  usually  has  a  face  value  of  $100, 
$500,  $50,  $1000,  etc.  A  share  of  mining  stock  often  has  a  much 
smaller  face  value. 

When  a  stock  company  pays  to  the  holders  of  its  stock  $2  on 
every  $100  of  its  capital  stock,  or  2%  on  its  stock,  the  company  is 
said  to  be  paying  a  $2  dividend,  or  a  2%  dividend. 

When  stock  is  paying  a  high  rate  of  dividend  it  may  sell  in  the 
market  above  par,  or  for  more  than  its  face  value.  If  the  rate  of 
dividend  is  low,  the  stock  may  sell  for  less  than  its  face  value,  or 
below  par.  When  it  sells  for  its  face  value  it  is  sold  at  par. 

A  Bond  is  a  written  agreement  made  by  a  national,  state,  or 
city  government,  or  by  a  company,  to  pay  the  holder  interest  at  a 
stated  rate  on  a  stated  sum  of  money,  called  the  face  of  the  bond. 

Stocks  and  Bonds  are  usually  purchased  through  an  agent, 
called  a  broker,  who  makes  a  business  of  buying  and  selling  stocks 
and  bonds.  He  charges  a  certain  per  cent  (usually  -J-%)  of  the 
face  value  for  buying,  and  an  equal  rate  for  selling.  This  charge 
is  called  brokerage. 

In  the  following  problems,  regard  the  face  value  of  the  stock 
or  bond  as  $100  and  the  brokerage  as  ^%,  unless  otherwise  stated. 

Following  is  a  list  of  newspaper  quotations  on  the  N.  Y.  stock 
market  on  certain  stocks  and  bonds: 


STOCKS 


Amal.  Copper 

American  Sugar 129% 

B.  &O.  com 10114 

B.  &  O.  pfd 95*4 

Chi.  &  Alton  com 36% 

Chi.  &  Alton  pfd 71* 

111.  Central  R.  R 148 

Peoples  Gas 105% 

Pressed  Steel  com ...  65* 

Pressed  Steel  pfd 94* 

Rubber  Goods  com . .  24* 

U.  S.  Steel  com 37% 

U.S.  Steel  pfd 87% 

St.  L.  and  S.  F.  com.  78* 
St.  L.  and  S.  F.  pfd. .  79% 


Open  High 
65%  66% 
129%  130 


95* 
37 

148% 
10575 
65* 


37% 
87i/2 
80% 


Low   Noon 
65%     65% 
139%  129% 
100%  101 
95*    95* 
861,4    36% 
71 K    72 
148      148% 
105*  105% 

94*  94* 

24*  24* 

37*  37* 

87%  87J4 

78  80% 

79%  80* 


BONDS 

Closing  bid  and  asked  prices  for  govern- 
ment bonds  were  as  follows: 

Bid  Asked 

New  2s  .......................  108%  109% 

Coupons  ........................  K'8%  109J4 

New  3s  ..........................  108^  109 

New  3s  coupon  ..................  108%  109 

New  3s  small   ..................  108  109 

Registered  4s  ..................  112*  113 

Coupon  4s  .......................  11214  113 

Registered  4s  new.  .  ............  139^  139% 

Coupon  4s  new  .................  139}i  139% 

Registered  5s  ...................  107%  107% 

Coupon  5s  .................  t  .....  107%  107% 


BOND  TRANSACTIONS 


No.  Sales 

27000  Atchison  Gen.  4s 100%@100^ 

12000  B.  and  O.  gold  4s 101    @101% 

1000  B.  and  O.  8i/2s 94i/2 

1000  C.  B.  and  Q.  4s 106% 

9000  C.  and  A.  3*s 76*@  76% 

25000  U.  P.  4s 


8514 


No.  Sales 

5000  U.  S.  Leather  4s 

3000  B.  &  O.  coupon  4s 
82000  Cen.  Pac.  R.  R.  3>/2s  .....  88M( 

10000  Wabash  R.  R.  5.s  ......  115K®115} 

10000  Manhattan  4s  ...........  105* 


1.  A  certain  express  company  has  a  capital  stock  of  $500,000, 
divided  among  its  stockholders  in  shares  of  the  par  value  of  $100 
each.     In  6  mo.  the  net  profits  amount  to  $10,500,  which  the 
company  distributes  among  its  stockholders  as  a  dividend.     What 
is  the  rate  of  dividend?     How  much  does  a  holder  of  200  shares 
receive  as  dividend  ? 

2.  What  must  a  man  have  paid  for  400  shares  Amal.  Copper 
stock  on  the  date  of  the  table,  including  brokerage,  if  he  bought 
at  the  opening  price?  at  the  highest  price?  at  the  lowest?  at  the 
noon  price? 

3.  Answer  similar  questions  for  IT.  S.  Steel  pfd.  (preferred). 

DEFINITIONS. — Preferred  stocks  are  stocks  which  pay  a  fixed  dividend 
(say  of  1%)  before  any  dividends  are  paid  on  common  stocks ;  Common 
stocks  pay  dividends  dependent  on  the  net  earnings  of  the  company 
after  expenses  and  dividends  on  preferred  stock  have  been  paid. 

4.  How  much  did  a  man  gain  or  lose  by  buying  2000  Am.  Sugar 
at  the  opening  price  and   selling  at  the  highest  price,  paying 
brokerage  of  £%  for  buying  and  also  for  selling? 

5.  Answer  similar  questions  for  B.  &  0.  com. ;  for  B.  &  0.  pfd. 

6.  How  much  does  a  man  make  or  lose  who  buvs  2500  shares 
111.   Cen.  R.  R.  stock  at  the  lowest  quotation  of  the  table  and 
sells  at  the  highest,  paying  brokerage  of  -J  for  buying  and  -J  for 
selling? 

7.  If  a  man  invests  in  St.  L.  and  S.  F.  pfd.  stock,  paying  7% 
annual  dividend,  at  the  lowest  quotation  of  the  table,  what  interest 
does  he  receive  on  his  investment? 

SUGGESTION. —The  investor  must  pay  $79*  -f-  $£  per  share  (of  $100), 
and  he  receives  $7  per  year  as  interest. 

8.  Answer  a  similar  question  on  C.  &  A.  pfd. 

9.  What  interest  does  a  man  receive  on  an  investment  in  Gov- 
ernment New  2s  if  he  invests  at  the  price  "bid,"  brokerage  £?  at 
the  price  "asked"? 

NOTE. — Government  2s,  8s  etc.,  are  government  bonds  paying  2%, 
3%,  etc.,  annually. 

10.  Answer  similar  questions  for  New  3s;  for  New  3s  cou- 
pon ;  for  Registered  4s ;  for  Registered  4s  new. 

11.  What  rate  of   interest   does  the  investor  who  bought  the 
1000  B.  &  0.  3As  receive? 


APPLICATIONS   OF    PERCENTAGE  347 

12.  Answer  similar  questions  for  the  investor  who  bought  the 
9000  C.  &  A.  3£s,  if  he  paid  the  lowest  quoted  price;  the  highest. 

13.  Which  pays  the  higher  rate  of  interest  on  the  investment, 
and  by  bow  much,  U.  P.  4s  as  quoted,   or  Wabash   E.  R.  5s  at 
the  lowest  quotation?  at  the  highest? 

14.  Which  is  the  more   profitable  investment,   and   by  how 
much,  B.  &  0.  coupon  4's  at  the  lowest  quotation,  or  a  straight 
loan  at  4%? 

15.  Answer  other  similar  questions  on  the  table. 

10.  An  investor  in  Chi.  &  Alton  com.  stock  at  the  highest 
quotation  received  two  2%  and  one  3%  dividend,  and  sold  the 
stock  18  mo.  later  at  38-J-.  What  rate  of  interest  did  he  receive 
on  his  investment?  How  much  profit  did  he  make  if  he  bought 
1000  shares? 

17.  Answer  similar  questions  on  the  table. 

§207.  Compound  Interest. 

With  certain  classes  of  notes,  if  the  interest  is  not  paid  when 
due,  it  is  added  to  the  principal  and  this  amount  becomes  a  new 
principal,  which  draws  interest.  Savings  banks  add  the  interest 
on  savings  deposits  at  each  interest-paying  period  and  pay  interest 
on  the  entire  amount.  This  interest  on  interest  is  called  com- 
pound interest.  Bankers  also  collect  their  interest  on  loans  and 
then  reloan  the  interest,  and  in  this  way  virtually  receive  com- 
pound interest  on  their  money. 

Compound  interest  is  usually  payable  annually,  or  semi- 
annually. 

1.  Find  the  compound  interest  on  a  note  of  $200  at  6%  for 
5  yr.,  interest  payable  annually. 

SOLUTION.— 

Interest  on  $200  for  1  yr.  at  6%  =  200  X  $  .06  =  $  12.00 
Amount  of  $200  for  1  yr.  at  6%  =  200  X  $  1.06  =  $212.00 
Amount  of  $212  for  1  yr.  at  §%  =  200  X  $(1.06)2  =  $224.72 
Amount  of  $224.72  for  1  yr.  at  Q%  =  200  X  $(1-06)3  =  $238.23 
Amount  of  $238.23  for  1  yr.  at  6%  =  200  X  $(1.06)*  =  $252  52 
Amount  of  $252.52  for  1  yr.  at  6%  =  200  X  $(1.06)5  =  $267.68 

The  compound  interest=$267. 68— $200=$67.68. 

NOTE. — The  $267.68  is  called  the  compound  amount.  The  expressions 
(1.06)3,  (1.06)*.  and  (1.06)5  mean  1.06  X  1.06  X  1.06,  1.06  X  1-06  X  1.06  X 
1.06,  and  1.06  X  1.06  X  1.06  X  1.06  X  1.06,  and  are  read  "1.06  cube," 
"1.06  to  the  4th  power,"  and  "1.06  to  the  5th  power." 


348  RATIONAL    GRAMMAR    SCHOOL    ARITHMETIC 

2.  Find  the  simple  interest  on  a  note  for  the  same  face  at  the 
same  rate  and  for  the  same  time  as  in  problem  1.     By  how  much 
does  the  compound  interest  exceed  the  simple  interest? 

3.  Find  the  compound  interest  on  $180  at  6%  for  3  yr.,  interest 
compounded  semi-annually. 

SOLUTION. — 

Amount  on  $180  for  J  yr.  at  6%  =  180  X  $  1.03  =  §185.40 
Amount  on  $185.40  for  £  yr.  at  6%  =  180  X  $(1.03)2  =  $190.96 
Amount  on  $190.96  for  £  yr.  at  Q%  =  180  X  $(1.03)3  =  $196.69 
Amount  on  $196.69  for  £  yr.  at  6%  =  180  X  §(1.03)*  =  $202.59 
Amount  on  $202.59  for  \  yr.  at  6%  =  180  X  $(1.03)5  =  $208.67 
Amount  on  $208.67  for  \  yr.  at  6%  =  180  X  $(1.03)6  =  $214.93 

Compound  interest  =  $214.93  —  $180  =  $34.93. 

4.  If  P  denotes  any  principal   at  •/•%  for  n  yr.,  interest  being 
compounded   annually,    show   that   if  A   denotes  the  compound 
amount, 


5.  Show  that  if  the  interest  is  compounded  semi-annually, 

(r    \2n 
1+<Too)  ' 

the  letters  meaning  the  same  as  in  Problem  4. 

6.  Show  that  if  /  denotes  the  interest,  compounded  annually, 


7.  If  the  interest  is  compounded   semi-annually,   show   that 

(r    \2n 
1+ *»)-*•   . 

8.  A  man  has  a  deposit  of  $100  in  a  savings  bank  paying  4%, 
interest  compounded  semi-annually,  for  3  yr.     How  much  is  due 
the  depositor  at  the  end  of  that  time? 

9.  How  much  is  due  on  a  savings  deposit  of  $250,  in  a  savings 
bank  paying  3%  semi-annually,  the  deposit  having  been  in  the 
bank  3^  years? 


USE    OF   LETTEKS    TO    REPRESENT   NUMBERS  349 

10.  A  man  paid  compound  interest  at  6% ,  interest  compounded 
annually,  on  a  note  of  $150  for  7  yr.     How  much  was  required  to 
pay  the  note? 

NOTE. — Many  notes  have  coupon  notes  attached  for  the  amount  of 
interest  due  at  each  interest-paying  period.  These  coupon  notes  usually 
bear  a  higher  rate  of  interest  than  does  the  note  itself. 

11.  A   note   for   $500,   bearing  6%   interest,    has   3   interest 
coupons  attached,  the  coupons,  when  due,  bearing  7%  interest, 
payable  annually.     Tf  the  note  runs  4  yr.,  no  coupons  being  paid 
in  the  meantime,  how  much  money  will  be  required  to  pay  the 
entire  debt  at  the  end  of  this  time? 

12.  A  note  of  8450,  bearing   interest   at   7%,  interest  pay- 
able  semi-annually,    coupons   bearing   8%   interest,    amounts   to 
what  sum  at  the  end  of  4£  yr.,  no  coupons  being  paid  until  the 
end  of  the  time? 


USE  OF  LETTERS  TO  REPRESENT  NUMBERS 

§208.  Problems. 

1.  Joseph  had  45$  and  he  earned  15$  more  selling  oranges. 
How  many  cents  had  he  then?     (Answer  by  indicating  the  opera- 
tion you  perform,  thus:  45$  +-  15$.) 

2.  William  had  x  marbles  and  Harold  gave  him  y  marbles. 
How  many  marbles  had  he  then? 

3.  A  cow  gave  x  Ib.  of  milk  at  the  morning  milking  and  y  Ib. 
at  the  evening  milking.     How  many  pounds  did  she  give  at  both? 

4.  James  earned  80$  selling  papers  on  Saturday  and  spent  45$ 
of   his  earnings.      How   many  cents  did  he  save  on   Saturday? 
(Answer  by  indicating  the  operation  you  use.) 

5.  During  July  a  boy  earned  m  dollars  and  spent  s  dollars. 
How  many  dollars  did  he  save? 

6.  Helen  bought  15  pencils  at  3$  apiece.     How  much  did  she 
pay  for  all? 

7.  Elizabeth  took  x  music  lessons  during  February  at  y  dollars 
per  lesson.     What  was  the  cost  of  her  lessons  for  the  month? 


30U  RATIONAL    GRAMMAR    SCHOOL    ARITHMETIC 

8.  A  thermometer  rose  A°  on  one  day  and  5  times  as  much 
the   day   following.     How   much   did    it   rise   on   the   following 
day? 

9.  The  area  of  a  rectangular  lot  is  63  sq.  rd.  and  the  length  of 
one  side  is  9  rd.     How  long  is  the  other  side? 

10.  The  area  of  a  rectangle  is  x  square  inches  and  the  length 
of  one  side  is  y  inches.     How  long  is  the  other  side? 

11.  How  many  lots  each  of  b  ft.  frontage  can  be  made  from  a 
frontage  of  a  feet? 

12.  An  orange  boy  earns  a  cents  on  Wednesday,  three  times 
as  many  cents  on  Thursday,  and  as  many  cents  on  Friday  as  on 
Wednesday  and  Thursday  together.     On  all  three  days  he  earns 
S0(f.     How  much  does  he  earn  on  Thursday?  on  Friday? 

NOTE. — In  all  such  problems  use  the  equation.  We  have  a  -f-  3a  -f-  4a 
=  80,  or  Sa  =  80.  If  8a  ==  80,  a  =  10.  3a  =  80,  Thursday  earnings,  and  4a 
=  40,  Friday  earnings. 

13.  The  altitude  of  a  rectangle  is  x  in.  and  the  base  is  3#  in. 
How  long  is  the  perimeter?     What  is  the  area  of  the  rectangle? 

14.  4«  means  4  x  a.     Compare  the  values  of  a  x  4  x  a  and  4  x 
a  x  a.    How  is  a  x  a  written?    How  is  4  x  a  x  a  written?    Ans.  4<r. 

15.  The  mercury  stood  at  x  degrees  at  2  p.m.  and  fell  3°  dur- 
ing the  next  hour.     The  reading  at  3  p.m.  was  25°.     What  was 
the  reading  at  2  p.m.? 

16.  The  mercury  stood  at  12°  at  8  a.m.     During  the  next  two 
hours  it  rose  x  degrees.     What  was  the  thermometer  reading  at 
10  a.m.?     If  it  had  fallen  y  degrees,  what  would  have  been  the 
reading  at  10  a.m.? 

17.  The  thermometer  read  28°,  the  mercury  fell  x  degrees, 
then  3x  degrees,  and  then  rose  6x  degrees,  when  the  reading  was 
32°.     What  was  the  number  of   degrees   in  each  of  the  three 
changes? 

18.  James  paid  $x  for  a  hat  and  twice  as  much  for  a  coat. 
He  paid  $4.50  for  both.     What  did  he  pay  for  each? 

19.  I  paid  $30  for  a  bicycle  and  sold  it  for  $x.     How  much 
did  I  gain? 

20.  I  paid  i  of  what  I  gained  for  a  coat.     How  much  did  I  pay 
for  the  coat? 


USE    OP   LETTERS   TO    REPRESENT   NUMBERS  351 

21.  Louis  had  x  apples  and  ate  y  of  them.     How  many  did  he 
have  left? 

22.  Henry  had  la  papers  and  sold  4a  of  them.     How  many 
did  he  have  left? 

23.  I  walked  x  miles  due  south  one  day  and  y  miles  due  north 
the  next  day.     How  far  was  I  then  from  my  starting  point? 

24.  A  man  rows  a  boat  downstream  at  a  rate  that  would  carry 
his  boat  a  miles  per  hour  through   still  water,  and  the  current 
alone  would  carry  him  down  b  miles  per  hour.     How  far  will  he 
go  in  1  hour? 

25.  A  man  walks  from  rear  to  front  through  a  railway  coach 
3  mi.  per  hour,  and  the  coach  is  running  at  the  same  time  a  mi. 
per  hour.     How  fast  does  the  man  pass  the  telegraph  poles  along 
the  track? 

26.  How  fast  does  he  pass  them  if  he  walks  from  front  to 
rear? 

27.  Mary  had  a  pencils  and  sold  them  at  50  apiece.     How 
many  cents  did  she  receive  for  them? 

28.  James  sold  x  oranges  at  a  cents  apiece.     How  many  cents 
did  he  receive  for  them? 

29.  A  dealer  sold  a  wagons  for  40«  dollars.     What  was  the 
price  per  wagon? 

30.  A  farmer  paid  x  dollars  for  15  A.  of  land.     How  much 
did  he  pay  per  acre? 

31.  The  area  of  a  rectangle  is  m  sq.  rd.  and  it  is  I  rd.  long. 
How  wide  is  the  rectangle? 

32.  A  township  is  x  mi.  square  and  contains  36  sq.  mi.     What 
is  the  value  of  .T?     What  does  x  represent? 

33.  Helen  bought  x  dolls  at  100  apiece  and  y  yd.  of  muslin  at 
80  a  yard.     How  many  cents  did  she  pay  for  both? 

34.  James  had  m  cents  and  earned  c  cents  more.     He  invested 
all  his  money  in  papers  at  30  apiece.     How  many  papers  did 
he  buy? 

35.  He  sold  his  papers  at  50  apiece.    How  much  did  he  receive 
for  the  papers?     How  much  did  he  gain? 

36.  The  base  of  a  triangle  is  x  ft.  and  the  altitude  is  y  ft. 
What  is  the  area  in  square  inches? 


352 


RATIONAL    GRAMMAR    SCHOOL    ARITHMETIC 


37.  The  area  of  a  triangle  is  x  sq.  ft.  and  the  base  is  6  ft.  long. 
What  is  the  altitude? 

38.  A  parallelogram  has  a  base  (x  +  ij]  in.  long,  and  is  6  in.  high. 
What  is  the  area?     Express  this  answer  in  two  ways  and  make  an 
equation  by  writing  the  two  expressions  equal.     Why  are  the  two 
expressions  equal? 


FIGURE 


209.  Uses  of  the  Equation. 

1.  If  each  brick  weighs  7  Ib.  and  there  are 
10  bricks  in  the  bucket  (Fig.  247),  with  how  many 
pounds  of  force  does  the  bucket  pull  downward 
on  the  rope,  the  bucket  itself  weighing  5  pounds? 

2.  If  we  denote  the  number  of  pounds  of  force 
with  which  the  man  pulls  downward  on  the  rope 
to   balance  the   bucket  by  p,  write  an  equation 
showing  the  number  of   pounds  in  p,  the  5-lb. 
bucket  being  loaded  with  10  bricks. 

3.  A    horse    is    raising     a 
10-ft.    steel    I-beam    weighing 
39  Ib.  for  each  foot  of  length. 

If  the  force  the  horse  must  exert  to  hold  the 
beam  suspended  in  the  air  be  denoted  by  F, 
write  an  equation  showing  the  number  of 
pounds  in  F. 

4.  A  wagon  weighing   1800   Ib.  is   loaded 
with  3  T.  of  coal.     When   it  is  being  drawn 

over  ordinary  pavement  it  pulls  backward  on  the  traces  of  the 
team  with  a  force  of  -fa  as  many  pounds  as  there  are  pounds  in 
the  entire  weight  of  the  coal  and  wagon.  If  the  force  exerted  by 
the  moving  team  is  ^Ib.,  write  an  equation  showing  the  number 
of  pounds  in  F. 

5.  Write  an  equation  showing  how  many  pounds  each  horse 
draws  if  both  draw  with  equal  force. 

These  problems  show  how  the  equation  may  be  used  in  solving 
simple  problems  of  mechanics  ;  and  we  need  to  learn  the  laws  upon 
which  the  use  of  the  equation  is  based. 


USE   OF   LETTERS   TO    REPRESENT    NUMBERS  353 

§210.  Principles  for  Using  the  Equation. 
Review  p.  103. 

1.  Two  weights,  one  of  x  lb.  and  the  other  of  y  lb.,  are  tied 
to  strings  which  pass  over   pulleys  at  A  and  B 

(Fig.  249).  The  strings  are  knotted  at  C.  If  x 
is  greater  than  ?/,  how  will  C  move?  Under  what 
condition  will  C  remain  at  rest?  What  relation 
is  shown  to  exist  between  x  and  y  by  a  move- 
ment of  C  toward  the  right? 

C"s  standing  stationary  indicates  a  balance  in  value  between 
the  forces  y  lb.,  drawing  toward  the  right,  and  x  lb.,  drawing 
toward  the  left.  This  fact  is  expressed  in  symbols  by  writing 
x  lb.  =  y  lb.,  or,  better,  by  x  =  y,  simply. 

2.  If  a  10-lb.  weight  be  hung  to  the  weight  ?/,  how  many  5-lb. 
weights  must  be  hung  to  x  to  secure  a  balance  of  value  ? 


NOTE.  —  This  is  shown  by  writing  y-\-W  =  sc-\-5-\-5,  read  "y  plus  10 
equals  x  plus  5  plus  5." 

3.  How  many  pounds  in  all  were  added  to  #? 

4.  If  50  lb.  were  added  on  the  right,  how  many  5-lb.  weights 
must  be  added  on  the  left  before  the  sign  of  equality  (=)  can  be 
written  between  the  numbers?     If  7  a  lb.  were  added  on  the  right, 
how  many  rt-lb.  weights  would  be  needed  on  the  left  for  balance? 

5.  How  many  pounds  would  be  needed  on  the  right  if  b  lb.  are 
added  on  the  left?  b  +  c  lb.?  p  +  35  pounds? 

6.  Write  the  equation  form  of  statement  for  each  case  of  Prob- 
lems 4  and  5. 

7.  If  a  greater  number  of  pounds  were  added  to  y  lb.  than  to 
x  lb.,  say  15  lb.  to  y  lb.  and  10  lb.  to  x  lb.,  what  would  occur  in 
the  apparatus  shown  in  Fig.  249? 

This  destroying  the  balance,  or  equality,  is  stated  briefly  in  signs  by 
writing  y  -f  15  >  x  -f-  10,  which  is  read  "?/  plus  15  is  greater  than  9  plus  10." 

If  the  balance,  or  equality,  is  destroyed  by  adding  a  heavier  weight  to 
x  than  to  y,  the  fact  is  stated  in  symbols  thus:  y  -f~  10  <  x  -\-  15  ;  read 
"y  plus  10  is  less  than  x  plus  15." 

Notice  in  each  case  that  the  vertex  of  the  horizontal  V  always  points 
toward  the  smaller  number. 

DEFINITIONS.  —  An  expression  in  which  the  sign  <  or  >  stands  between 
two  numbers  is  called  an  expression  of  inequality. 


354  RATIONAL   GRAMMAR   SCHOOL   ARITHMETIC 

We  now  have  signs  for  writing  briefly  the  three  possible  relations 
which  may  exist  between  any  two  numbers,  as  a  and  b.  They  are  called 
relation  signs. 

8.  Bead  and  give  the  meanings  of    (1)  a  >  b,  (2)  a  =  b,  (3) 
a  <  b,  (4)  x  +  20  <  x  4- 15  -f  10. 

9.  If  y  =  x,  and  a  Ib.  are  added  to  y  lb.  and  b  lb.  to  x  lb.,  how 
will  the  knot  G  move  for  case  (1)  Problem  8?  for  case  (2)?  for 
case  (3)  ?     State  in  symbols  the  fact  shown  by  the  knot  C  in  each 
case  after  adding  the  weights,  using  the  correct  relation  signs. 

10.  If  any  given  weight,  a  lb.,  be  added  to  y,  what  must  be 
true  of   the  weight  b  lb.  if,   when  added  to  x,  we  may  write 
y  +  a  =  x  +  b? 

11.  If  z  lb.  be  added  to  both  the  y  lb.  and  the  x  lb.,  when 
y  -  x,  what  equation  may  we  write? 

PRINCIPLE  I  (FOR  ADDITION  OF  EQUATIONS). — If  the  same 
number,  or  equal  numbers,  be  added  to  both  sides  of  an  equation, 
the  sums  are  equal. 

ILLUSTRATION. — If  y  =  x,  and  a  =  b,  then  we  may  write 

y  -\-  a  =  x  -\-  a, 
and  y  -\-  b  =  x  -\-b, 
and  y  -\-  a  =  x  -\-  b, 
and  y  -\-  b  =  x  +  a. 

12.  If  8  lb.  be  removed  from  the  left  scale  pan  (Fig.  249),  how 
many  4-lb.  weights  must  be  removed  from  the  right  pan  to  restore 
the  balance  ? 

NOTE. — This  is  written  in  signs  y  —  8  =  x  —  4  —  4,  and  is  read  "y  —  8 
equals  x  minus  4  minus  4." 

13.  How  many  pounds  in  all  were  removed  from  the  right 
scale  pan? 

14.  If  y  =  x,  and  if  a  lb.  be  removed  from  the  left,  and  b  from 
the  right,  to  what  must  the  value  of  b  be  equal  to  enable  us  to 
write  y  —  a  =  x  —  b? 

PRINCIPLE  II  (FOR  THE  SUBTRACTION  or  EQUATIONS). — If  the 
same  numjter,  or  equal  members,  are  subtracted  from  equal  num- 
bers, the  differences  are  equal. 

ILLUSTRATION.— If  y  =  x  and  a  =  b,  then  we  may  write: 

y  —  a  =  x  —  a, 
and  y  —  b  =  x  —  b, 
and  y  —  a  =  x  —  b, 
and  y  —  6=0?  —  a. 


USE   OF   LETTERS   TO    REPRESENT   NUMBERS  355 

15.  If  in  Fig.  249  y  be  doubled,  what  corresponding  change  in 
x  will  restore  the  balance? 

NOTE.— The  double  of  y  is  written  2y  and  read  "two  ?/." 

16.  What  equation  states  that  there  is  a  balance? 

17.  If  y  =  x,  and  if  4  weights,  each  equal  to  y,  are  put  on  the 
right  of  the  apparatus  in  Fig.  249,  how  many  weights.,  each  equal 
to  x,  must  be  put  on  the  left  to  keep  the  balance? 

18.  If  a  weights,  each  equal  to  y,  are  on  the  right,  how  many 
weights,  each  equal  to  x,  must  go  on  the  left  to  secure  balance? 

NOTE. — The  equation  is  ay  =  ax. 

19.  If  a  =  5,  and  a  weights,  each  equal  to  y,  are  put  on  the 
right,  and  I  weights,  each  equal  to  #,  are  .put  on  the  left,  what 
will  be  shown  by  the  scales  if  y  =  x.     (Answer  with  an  equation.) 

PRINCIPLE  III  (FOR  THE  MULTIPLICATION  OF  EQUATIONS).— 
If  equal  numbers  are  multiplied  by  the  same  number,  or  by  equal 
numbers,  the  products  are  equal. 

ILLUSTRATION. —If  y  =  x  and  a  —  b,  then 

ay  =  ax, 
and  by  =  bx, 
and  ay  =  bx, 
and  ax  =  by. 

20.  If  y  =  x,  and  half  of  the  weight  on  the  left  be  taken  off, 
what  fractional  part  of  the  weight  on  the  right  must  be  taken  off 
to  restore  the  balance?  * 

21.  If  only  £  of  y  be  kept  on  the  right  (Fig.  249),  what  frac- 
tional part  of  x  must  remain  on  the  left? 

22.  If  only  ^  Ib.  be  kept  on  the  right  (Fig.  249),  what  frac- 
tional part  of  x  Ib.  must  remain  on  the  left  for  balance? 

23.  Suppose  y  =  x  and  a  =  #,  and  that  —  Ib.  are  on  the  right, 

x 
and  -7  Ib.  are  on  the  left;  what  equation  would  be  shown  to  be 

true  by  the  apparatus? 

NOTE. — The  equation  is  —  =  -r,  read  "y  divided  by  a  equals  x  divided 
by  6." 


356  RATIONAL   GRAMMAR   SCHOOL   ARITHMETIC 

PRINCIPLE  IV  (FOR  THE  DIVISION  OF  EQUATIONS).  —  If  equal 
numbers  are  divided  ~by  the  same  number,  or  by  equal  numb&rs, 
the  quotients  are  equal. 

ILLUSTRATION.—  If  y  ==  x  and  a  =  b,  then 


and      =, 
b       b 


b 

and|=*. 
6       a 

In  Fig.  183,  p.  297,  a  (x  +  y)  represents  the  area  of  the  whole 
rectangle,  while  ax  and  ay  represent  the  areas  of  its  two  parts. 
As  the  two  parts  of  the  large  rectangle,  taken  together,  must 
equal  the  whole  rectangle,  we  may  write  : 

a  (x  +  y)  =  ax  +  ay 

In  Fig.  184,  p.  298,  the  area  of  the  whole  rectangle  is  given 
by  (a  +  b)  (x  -f  ?/),  while  the  areas  of  the  several  parts  are  ax,  ay, 
bx,  and  by.  Since  the  parts,  taken  together,  make  up  the  whole 
rectangle,  we  may  write: 

(a  -t-  b)  (x  +  y)  =  ax  +  ay  +  bx  +  by 

These  two  equations  illustrate  the  meaning  of  a  fifth  principle 
of  the  equation,  viz.  : 

PRINCIPLE  V.  —  Any  whole  equals  the  sum  of  all  its  parts. 

These  five  fundamental  principles  or  laws,  exemplified  by  the 
scales,  p.  103,  and  the  pulley  device  (Fig.  249),  p.  353,  must  not 
be  violated  in  using  the  equation. 

§211.  Problems. 

1.  Find   the   value  of  the.  letter  #,  y,  or  z,  in  each  of  the 
first  nine  problems: 

(1)  3x  =  15  Ib.  ;  (4)  9*  =  36<p  ;  (7)  llx  =  55  ; 

(2)  Sx  =  24  ft.  ;  (5)  f  x  =  $3  ;  (8)    Jy  =  21  ; 

(3)  ly  =  18  mi.  ;  (6)  \y  =  14  sq.  in.  ;     (9)    ax  =  3a. 

2.  If  3x  =  9,  to  what  is  2s:  equal?  7x?  %x? 

3.  If  Ix  =  21,  to  what  is  5x  equal?  fz? 


USE    OF    LETTERS   TO    REPRESENT   NUMBERS  357 

4.  If  x  +  3  =  8,  what  is  x?  3z?  5x?  x2? 

5.  If  2x  +  7  =  15,  what  is  a?  3z?  7a?  _z2?  \/z? 
G.   If  #-3  =  6,  what  is  a;?  3z?  a;2?  \/a7? 

7.  2:i;  +  3a  +  Qx  =  33  ;  find  x. 

8.  4z-2#  +  £  =  64;  find  a;. 

9.  7x-x  +  2x  =  32;  find  #. 

10.  xy  =  16,  and  ?/  =  8  ;  what  is  a;?     If  a;  =  8,  what  is  #? 

11.  #£  =  35,  and  a  =  5  ;  what  is  a;?     If  x  =  5,  what  is  a? 

12.  «J  =  10;  what  is  b  if  a  =  1?  2?  3?  4?  10?  20? 

13.  By  what  principle  may  we  write 

c(x  +  y  -4-  z)  =  ex  +  cy  +  cz, 

x,  y,  and  z  denoting  the  bases  of  3  rectangles  whose  altitudes  are 
each  equal  to  c?  (Answer  by  sketching  the  proper  figure  and 
pointing  out  the  rectangles  whose  areas  represent  each  of  the 
products  in  the  equation.) 

14.  If  a  =  8  and  5  =  5,  find  the  values  of  the  following  expres- 
sions and  tell  what  ones  are  equal  : 


(1)  (rt  +  fc);      (4)  («-f6)2;      (?)  «s-2oi  +  J8;      (10)  «3  +  52; 

(2)  (a  -  b)  ;      (5)  (a  -  b)'2  ;      (8)  a2  +  2ab  +  W  ;       (11)  a(a*  +  V  ; 

(3)  2a&;         (6)«2  +  62;        (9)  (a  +  V)(a-V)\     (I2)b(a-b). 


15.  If  a:  =  9  and  y  =  4,  tell  which  of  the  following  express  true 
relations  and  write  the  correct  relation  sign  in  each  case  :  (See  p. 
354.) 


(12)  (x- 


(4)y<a;  (14)  (a:- 

(5)  ^  +  ^  =  13;  (15)  (^- 

(6)  *-jr-A;  (i6)  (»+ 

(7)  (a;  +  ?/)2  =  169;  (17) 

(8)(a:-y)^25;  (18)  (a:- 


(10)  a;2  +  /  =  169  ;  (20)  x  (x  +  y)=  x2  +  a;y. 


358  RATIONAL   GRAMMAR    SCHOOL   ARITHMETIC 

§212.  Statements  in  Words  and  in  Symbols. 

1.  Write  the  symbolic  statements  for  these  verbal  phrases  and 
statements.     Let  x  stand  for  the  number  when  there  is  but  one 
number  to  be  symbolized  in  the  problem. 

(1)  A  certain  number  increased  by  15. 

(2)  Twice  a  number  diminished  by  8. 

(3)  Seven   times   a   number    increased    by   three    times   the 
number. 

(4)  The  square  of  a  number,  divided  by  8. 

(5)  The  sum  of  the  square  and  the  first  power  of  a  number. 

(6)  Eight  times  a  number,  divided  by  three. 

(7)  One-third  of  10  times  a  number. 

(8)  Three  times  a  certain  number,  diminished  by  one,  equals  20. 

(9)  Eighteen  times  the  square  of  a  number  equals  72. 

(10)  Twenty-five  times  a  number,  increased  by  5,  equals  30 
times  the  number,  diminished  by  15. 

(11)  One-eighth  of  the  sum  of  a  certain  number  and  18. 

(12)  Six  times  the  difference  between  a  certain  number  and 
3  equals  18. 

(13)  The  product  of  the  sum  and  the  difference  of  x  and  3 
equals  18  (x  being  greater  than  3). 

(14)  The  difference  between  x  and  18  is  greater  than  18;  is 
less  than  25;  is  equal  to  20  (x  >  18  in  each  case). 

2.  State  in  words  what  these  expressions  mean.     For  example, 
(1)  means  "double  a  certain  number,  diminished  by  9": 

(1)  33-9;  (6)    (x  +  l)  (»  +  !);       (11)2^  +  6  =  12; 

(2)  16»  +  a;;  (7)  *(&-4);  (12)   7z-2  +  16; 

(3)  282; +  17;  (8)  3(9 -a);  (13)   5z+7  =  42; 

(4)  z*  +  x\  (9)  12(^2-1);  (14)  (a - 4)  (z  +  4)  =20; 

(5)  xz-x^  (10)  (a-1)  (z-1);  (15)  (a  +  b)(a-b)  =  a*-b\ 

3.  Translate  into  symbols  these  verbal  phrases  and  statements, 
using  a  and  &,  or  x  and  y,  for  the  two  numbers: 

(1)  The  sum  of  two  numbers  equals  25. 

(2)  The  difference  of  two  numbers  equals  15. 

(3)  The  sum  of  the  squares  of  two  numbers  is  less  than  27. 

(4)  The  square  of  the  sum  of  two  numbers  equals  100. 


USE    OF    LETTERS   TO    REPRESENT    NUMBERS  359 

(5)  The  difference  of  the  squares  of  two  numbers  equals  9. 

(6)  The  sum  of  the  squares  of  two  numbers  equals  seven  times 
the  difference  of  the  numbers. 

(7)  The  product  of  two  numbers  equals  their  sum. 

(8)  The  quotient  of  two  numbers  equals  their  difference. 

(9)  A  certain  number  increased  by  1  equals  another  number 
diminished  by  3. 

4.  Translate  into  words  these  symbolic  expressions.  For 
example,  (1)  means  "one-ninth  of  the  difference  between  6  times 
a  certain  number  and  its  square": 


"  y 

(3)    3-^;             (8)  £=£  =a  +  6;  (13) 

4                                 9*.-£;  (14) 

y      x  '  ' 


5.  Find  the  number  which  may  be  put  in  place  of  the  letter 
in  each  of  these  equations  to  furnish  true  equations: 
(1)  2®  +  3  =  7; 
SOLUTION.— 


-3 


2x        =  4    by  Principle  II, 

2ic_4_  i 

2  ~2 

x=  2    by  Principle  IV. 

Check :  205  +  3  =  2x2-1-3  =  44-3  =  7,  which  is  correct. 
(2)3^-1=5;         (4)t»  +  l  =  2;         (6)    6|  +  f  =  V4  5 
(3)  8a-6  =  18;         (5)  -3|--2  =  1;         (7)  -6f-f=V6. 
NOTE.— First  multiply  both  sides  of  (7)  by  21. 

6.  Find  the  value  of  the  numbers  of  problem  1  (8),  (9),  and 
(10). 


RATIONAL    GRAMMAR   SCHOOL   ARITHMETIC 

§213.  Problems. — FOR  EITHER  ARITHMETIC  OR  ALGEBRA. 

1.  The  mercury  column  in  a  thermometer  rose  a  certain  num- 
ber of  degrees  one  day,  and  3  times  as  many  degrees  the  next  day. 
It  rose  12°  during  the  2  days.     How  many  degrees  did  it  rise 
each  day? 

ARITHMETICAL  SOLUTION.— 

A  certain  number  denotes  the  rise  the  first  day. 

3  times  this  number  denotes  the  rise  the  second  day. 
Hence  4  times  a  certain  number  denotes  the  rise  in  two  days. 

4  times  a  certain  number  equals  12°  (by  the  given  problem). 
Once  the  number  equals  3  ,  the  rise  the  first  day  (Principle  IV). 

3  times  the  number  equals  9°,  the  rise  the  second  day  (Principle  III). 
Check    3°  +  9°  =  12°,  the  rise  in  two  days. 

ALGEBRAIC  SOLUTION.  — 

Let  x  denote  the  first  day's  rise. 
Then,  3.t  denotes  the  second  day's  rise. 
x  -\-  3x  denotes  the  rise  in  2  days. 
4x  =  12°. 

x  =  3°,  the  first  day's  rise  (Principle  IV). 
3x  =  9°,  the  second  day's  rise  (Principle  III). 
Check:  3°  -f  9°  =  12°. 

2.  A  man  bought  4  times  as  many  hogs  as  cows,  and  after 
selling  5  hogs  he  had  23  hogs  left.     How  many  cows  did  he  buy? 

ARITHMETICAL  SOLUTION.— 

A  certain  number  represents  the  number  of  cows  bought. 

4  times  this  number  represents  the  number  of  hogs  bought. 

4  times  this  number  minus  5  denotes  the  number  of  hogs  left. 

Then  4  times  this  number,  minus  5,  equals  23  (by  the  problem). 

4  times  this  number  equals  23  plus  5  (Principle  I). 

4  times  this  number  =  28. 

This  number  =  7,  the  number  of  cows  (Principle  IV). 

Check:  4  X  7  —  5  =  23. 
ALGEBRAIC  SOLUTION. — 

Let  x  denote  the  number  of  cows  bought. 

Then,  4x  denotes  the  number  of  hogs  bought. 

4x  —  5  denotes  the  number  of  hogs  left. 

Then  4x  —  5  =  23  (by  the  problem). 

4x  =  28  (Principle  1). 

x  =  7  (Principle  IV).     Ans.  7  cows. 

Check:  4  X  7  —  5  =  23. 

3.  Two  masses  were  placed  on  one  scale  pan  of  a  balance  and 
found  to  weigh  18  Ib.     One  of  the  masses  was  then  placed  in  each 
pan,  and  it  required  4  Ib.  additional  on  the  light  pan  to  balance 
the  scales.     What  was  the  weight  of  each  mass? 


USE    OF    LETTERS   TO    REPRESENT   NUMBERS  361 

ARITHMETICAL  SOLUTION.— 

A  certain  number  of  pounds  denotes  the  weight  of  the  heavier  mass. 

Another  number  of  pounds  denotes  the, weight  of  the  lighter  mass. 

The  first  number  plus  the  second  number  denotes  the  combined 
weight  (18  pounds). 

The  first  number  minus  the  second  denotes  the  difference  of  the 
weights,  or  the  additional  weight,  which  equals  4  pounds. 

2  times  the  first  number  equals  18  plus  4  equals  22. 

The  first  number  equals  11. 

11  plus  the  second  number  =  18.     (Principle  IV). 

The  second  number  =  7.     (Principle  II). 

The  weights  are,  then,  7  Ib.  and  11  Ib. 

Check:  11  Ib.  +  7  Ib.  =  18  Ib. 
11  Ib.  —  7  Ib.  =    4  Ib. 

ALGEBRAIC  SOLUTION. — 

Let  x  denote  the  number  of  pounds  in  the  weight  of  the  heavier  mass. 
Let  y  denote  the  number  of  pounds  in  the  weight  of  the  lighter  mass. 
Then,  z-f-  y  denotes  the  number  of  pounds  in  the  combined  weight,=l& 
x  —  y  denotes  the  number  of  pounds  in  the  additional  weight,  =  4. 

x  +  y  =  18 
x  —  y=    4 

2x          =  22  (Principle  I). 
x          =11  (Principle  IV). 

11  -f  y  =  18 

y  =  18  —  11  =•=  7  (Principle  II). 

Check:  11  -f  7  =  18 
11  —  7=    4 

Solve  the  following  problems  by  both  the  algebraic  and  tho 
arithmetical  method,  and  state  which  of  the  solutions  is  the 
shorter : 

4.  A  man  cut  for  me  3  times  as  many  ash  trees  as  oak  trees, 
and  as  many  hickory  trees  as  ash  trees  and  oak  trees  together. 
In  all  he  cut  32  trees.     How  many  of  each  kind  did  he  cut? 

5.  A  man  bought  4  times  as  many  2(f  stamps  as  5^  stamps, 
and  •§-  as  many  10$  stamps  as  %<f  stamps.     For  all  he  paid  $4.95. 
How  many  of  each  kind  did  he  buy? 

NOTE. — Let  x  denote  the  number  of  5^  stamps  purchased. 

6.  The  side  of  a  rectangular  field  is  twice  its  breadth  and  the 
distance  around  the  field  is  240  rd.     How  many  acres  does  the 
field  contain? 

7.  I  bought  a  number  of  books  one  day,  4  times  as  many  the 
next  day,  and  3  books  the  third  day.     In  all  I  bought  33  books. 
How  many  did  I  buy  eacli  clay? 


362  RATIONAL    GRAMMAR    SCHOOL    ARITHMETIC 

8.  A  newsboy  sold  a  certain  number  of  papers  on  Wednesday, 
twice  as  many  on  Thursday,   and   3   more   on   Friday   than  on 
Thursday.     In  all  he  sold   48  papers.      How  many  did   he  sell 
each  day? 

NOTE. — Let  x  denote  the  number  of  papers  sold  on  Wednesday. 

9.  Texas  lacks  17,470  sq.  mi.  of  being  large  enough  to  make 
5  states  the  size  of  Illinois.     How  large  is  Illinois  if  Texas  con- 
tains 265,780  square  miles? 

10.  Maude  has  78$,  which  lacks  18$  of  being  6  times  as  much 
money  as  James  has.     How  many  cents  has  James? 

11.  It  is  twice  as  far  from  Chicago  to  Niles  by  a  certain  route 
as  from  Niles  to  Albion,  and  from  Chicago  to  Albion  is  190  mi. 
How  far  is  it  from  Chicago  to  Niles? 

12.  Rochester,  N.  Y.,  is  17  mi.  more  than  twice  as  far  from 
Chicago  as  is  Detroit.     Rochester  is  587  mi.  from  Chicago.     How 
far  is  Detroit  from  Chicago? 

13.  Niagara  Falls  is  8  mi.  less  than  9  times  as  far  from  Chicago, 
as  is  Michigan  City.     Magara  Falls  is   514  mi.  from   Chicago. 
How  far  is  it  from  Chicago  to  Michigan  City? 

14.  It  is  4  times  as  far,  less  16  mi.,  from  Chicago  to  N.  Y. 
City  as  from  Chicago  to  Ann  Arbor.    If  it  is  976  mi.  from  Chicago 
to  N.  Y.  City,  how  far  is  it  to  Ann  Arbor? 

15.  Champaign  is  5  mi.  more  than  %  as  far  from  Chicago  as 
is  Cairo.     If  it   is   128  mi.   from  Chicago   to  Champaign,  how 
far  is  it  from  Chicago  to  Cairo? 

16.  A  father  and  his  son  together  earn  $75  a  month,  and  the 
father  earns  4  times  as  much  as  the  son.     How  much  does  each 
earn? 

17.  William  has  3  times  as  many  marbles  as  Joseph,  and  both 
together  have  24.     How  many  marbles  has  each? 

18.  A  line  32  in.  long  is  to  be  divided  into  2  parts,  one  of 
which  is  8  in.  longer  than  the  other.     How  long  is  each  part? 

19.  A  tree  75  ft.  high  was  broken  by  the  wind  so  that  the  part 
standing  was  15  ft.  longer  than  the  part  broken  off.     How  long 
was  each  part? 

20.  The  area  of  a  rectangular  field  is  6400  sq.   rd.  and  its 
lengtfy  is  160  rd. ;  what  is  its  width? 


USE   OF    LETTERS    TO    REPRESENT   NUMBERS  363 

21.  The  area  of  a  rectangular  sidewalk  is  120  sq.  rd.  and  the 
width  is  1|  yd.  ;  how  long  is  the  walk? 

22.  A  man  received  1180  for  72  days'  work;    what  was  his 
daily  wage? 

23.  Find  the  side  of  a  square  field  equal  in  area  to  a  rectangle 
80  rd.  long  by  20  rd.  wide. 

24.  A  man  received  13000  for  80  A.  of  land  ;   how  much  did 
he  receive  per  acre? 

25.  The  circumference  of   a  circle  is  2200  rd.  ;   what  is  the 
radius?     (Equation:  Gf  r  =.  2200.) 

2G.  The  area  of  a  circle  is  201^  m.2;  how  long  is  the  radius? 
the  circumference? 

27.  Find  the  radius  and  the  circumference  of  a  circle  whose 
area  is  1  square  inch. 

§214.  Formal  Work. 

Find  the  value  of  the  unknown  quantity,  x  or  y,  in  each  of  the 
following  : 

1.  Qx  -t-  4  =  22.  NOTE.—  First  multiply  both  mem- 

2.  3*-  10  =  14. 

3.  4</-2.?/  +  2  =  16.  13.  |5  =  5. 

4.  8#  +  y+3  =  21. 

5.  fa-  2  =  16.  u.  . 
6. 


16.  21  -a  =  33  -2z. 
NOTE.  —  First  multiply  both  mem- 

bers by  3.  NOTE.—  First  add  2x  to  both  mem- 

or  _  I  j.  bers,   then    subtract    21    from    both 

g  _  2  members. 

'  '  =*.+  90. 


10.  6(x  +  2)=  72.  18-  ^  ~  T  =  2X  +  39< 


14.  -      .1.5. 

X 


364  RATIONAL   GRAMMAR   SCHOOL   ARITHMETIC 

21.  In  the  following  equations  W  denotes  the  weight  of  the 
rope,  or  cable,  in  Ib.  per  yd. ;  L  the  working  load  in  tons;  S,  the 
breaking  load  in  tons,  and  7),  the  diameter  of  the  rope,  or  cable, 
in  inches. 

For  hemp  cable,  TF=.5777)2;  £  =  .1097)2;  S=  .654T)2. 

For  tarred  hemp  cable,  W=  1.0367)2;  L  =  .247T)2;  S=  1.4807)2. 

For  manila  rope,  W=.765D°~;  Z  =  .3297)2;  #=  1.8777)2. 

For  iron  wire  rope,   IF=  3.8477)'-;  7,  =  2.8627>2;  S=  17.0127)2. 

For  steel  wire  rope,  JF=  3.940/.>a;  £  =  4.441Z>2;  #=27.6307>2. 

Find  the  weight  per  yd.,  the  working  load  and  the  breaking 
load  of  a  hemp  cable  of  1"  diameter ;  of  2-J-"  diameter. 

22.  Solve  similar  problems  for  tarred  hemp  rope ;  for  manila 
rope. 

23.  Find  W,  L,  and  8  for  an  iron  wire  rope  for  which  D  =  1 J" ; 
D  =  2i". 

24.  For  the  same  values  of  7),  find  TF,  Z,  and  £  for  steel  wire 
rope. 


USES  OF  THE  EQUATION 

§215.  Sliding  and  Static  Friction. 

EXPERIMENT  I. — With  appliances  arranged  as  suggested  in  the 
cut,  the  empty  carriage,  B,  was  weighed  and  placed  upon  a  smooth 
«  surface  of  the  substance  whose  friction  was 

JH,  to  be  studied.     Known  weights  were  then 

placed  upon  the  pan  8  (which  was  also 
weighed)  until,  after  pushing  the  block 
loose  from  the  surface,  it  would  just  slide 
slowly  along  it.  Known  loads  were  then 
placed,  one  after  another,  upon  the  block 


at  7/,  and,  for  each  load,  weights  were 
put  upon  8  until  the  carriage  would  just  slide  along  after  being 
started.  The  loads  put  at  L  are  tabulated  in  the  first  column  below, 
and  the  weights  at  8  in  the  second.  The  weight  needed  at  8  in 
each  case,  together  with  the  weight  of  the  pan,  is  the  force  of 
friction,  or  friction,  simply. 

EXPERIMENT  II. — In  the  above  experiment  it  is  found  that  the 
surfaces  of  carriage-block  and  substance  cling  together  at  starting, 
and  that  a  slight  thrust  on  the  block  is  needed  to  start  it  to  avoid 


USES   OF   THE    EQUATION 


365 


overloading  at  S.  When  the  weight  at  S  is  heavy  enough  of  its 
own  account  to  start  the  load,  it  moves  off  with  increasing  speed; 
and  for  Experiment  I  this  must  not  be  allowed.  This  is  because 
friction  at  starting  is  greater  than  after  motion  has  begun.  The 
first  is  called  static  (standing)  friction,  and  the  second,  sliding 
friction.  The  static  friction  was  also  found  for  this  apparatus, 
and  is  tabulated  in  the  third  column  for  each  load. 


LOAD  INCLUD- 

FORCE OF  FRICTION  IN  LB. 

ING  CARRIAGE 

IN  LB. 

Sliding 

Static 

7.6 

1.0 

3.2 

11.6 

1.7 

4.4 

14.6 

2.0 

5.7 

18.6 

2.8 

6.8 

21.6 

3.0 

7.5 

25.6 

3.6 

9.2 

28.6 

4.0 

10.0 

"35.6 

4.9 

12.2 

42.6 

5.9 

14.7 

49.6 

6.9 

16.8 

566 

7.8 

20.0 

Starting  at  any  point  as  0,  Fig.  251,  on  a  page  of  cross-ruled 
paper,  draw  through  0  a  horizontal  line,  and  also  a  line  perpendic- 
ular to  the  horizontal.  Letting  a  horizontal  side  of  one  of  the  small 
squares  represent  5  lb.,  measure  off  from  0,  toward  the  right,  dis- 
tances to  represent  the  numbers  standing  in  the  first  column  of  the 


in   ., 

J 

1 

2  5 

z 

0 

P  2)4 

H     i 

J 

I 

B   ( 

:    I 

)E 

>® 

I<v 

y 

ct 

"•  n 

A 

A  < 

© 
,a 

,0       4 

,b  , 

i,e 

rf      rl 

h 

• 

J 

,i 

20       E5       30       35       40       45       50       55 
LOADS  IN  LB5. 
FIGURE  251 


table,  as  Oa,  Ob,  Oc,  etc.,  to  01.  Lay  off  the  tenths  by  careful 
estimate.  Then  letting  a  vertical  side  of  a  square  denote  2£  Ib.s., 
measure  vertically  upward  from  rt,  b,  c,  etc.,  to  A,  B,  C,  and  so  on 
distances  to  represent  to  this  scale  the  values  given  in  the  second 
column,  which  correspond  to  the  numbers  represented  by  the  lines 


366  RATIONAL   GRAMMAR   SCHOOL   ARITHMETIC 

0«,  Ob,  etc.  Denote  the  top  points  of  these  vertical  lines  by 
points  as  shown  in  Fig.  251,  /I,  /?,  C,  and  so  on,  to  L.  Place 
a  ruler,  or  other  straight  edge,  along  these  points.  What  do  you 
notice?  Do  the  points  seem  to  be  situated  in  random  positions, 
or  do  they  seem  to  be  located  in  accordance  with  some  law?  This 
law  may  be  called  the  straight  line  law. 

Calling  the  combined  weight  of  the  load  and  carriage  Z,  and  the 
force  of  sliding  friction  F,  the  ratio,  to  3  decimal  places,  of  F  to  L 

for  the  first  values  of  the  foregoing  table  (1:7.6)  is  .132  and  for 

-p 

the  last  values  (7.8:56.6),  -=-  =  .138.  The  mean  value  is  .135.  Since 

x/ 

F 

Y=  .135,  we  have  approximately, 

(I)  F=.IZ5L.     (Principle  III). 

From  equation  (I)  compute  to  2  decimals  the  value  of  F  for 
each  value  of  L  and  arrange  the  values  of  L  and  ^in  a  table  as 
shown  below.  The  measured  values  of  Fare  given  in  the  third 
column,  the  differences  of  the  measured  and  computed  values  of 
F  are  given  in  the  fourth  column.  The  sign  +  is  written  before 
the  difference  whenever  the  measured  jF  exceeds  the  computed  F. 
In  the  opposite  case  the  sign  —  is  used.  Columns  5  and  6  are 
explained  below. 

L  COMP'T'D  F     MEAS'D  F         DIFF.     2ND  COMP'T'D  F        DIFF. 

7.0  1.03  1.0  -.03  1.16  -.16 

11.6  1.56  1.7  +.14  1.69  +.01 

14.6  1.97  2.0  +.03  2.10  -.10 

18.6  2.51  2.8  +.29  2.04  +.16 

21.6  2.92  3.0  +.08  3.05  -.05 

25.6  3.45  3.6  +.15  3.58  +.02 

28.6  3.86  4.0  +.14  3.99  +.01 

35.6  4.80  4.9  +.10  4.93  -03 

42.6  5.75  .          5.9  +.15  5.88  +.02 

49.6  6.69  6.9  +.21  6.82  +.08 

50.0  7.64  7.8  +.16  7.77  +.03 

Add  all  the  +  differences,  from  the  sum  subtract  the  -  differ- 
ence and  divide  the  result  by  the  number  of  differences  (11). 
The  quotient  is  .13.  This  is  called  the  average  of  the  +  and  - 
differences  and  the  result  indicates  that  the  equation  should  give 
each  value  of  /"about  .13  larger  than  the  /"given  by  equation  (I). 
Accordingly  we  add  .13  to  the  second  member  of  equation  (I), 
making  the  equation  - 

(II)  F 


USES    OF   THE    EQUATION  367 

Substituting  the  values  of  L  in  column  1  successively  in  equa- 
tion (II),  we  obtain  the  values  of  Fin  column  5.  The  differences 
between  these  computed  values  of  J^and  the  measured  values  of 
column  3  are  given  in  column  G  as  before.  Using  common  frac- 
tions instead  of  decimals,  it  is  sufficiently  accurate  to  call  the  cor- 
rect law  of  sliding  friction: 


This  means  that  the  force  F,  in  lb.,  needed  to  overcome  slid- 
ing friction  of  the  surfaces  used,  equals  -f^  of  the  combined  load, 
L,  plus  4  of  a  pound. 

Substituting  successively  in  (III)  for  Z,  1,  10,  15,  20,  30,  40, 
50,  the  following  values  of  Fare  found,  0.3,  1.5,  2.1,  2.8,  4.1,  5.5 
and  6.8.  Plotting  these  values  of  L  horizontally  as  in  Fig.  251 
and  the  corresponding  values  of  ^vertically  the  points  indicated 
thus  A  are  obtained.  Do  they  seem  to  lie  along  the  same  line 
as  do  the  observation  points,  marked  thus  0? 

1  .  Similarly  the  equation  form  of  the  law  for  static  friction 
could  be  readily  found.  This  is  left  as  an  exercise  for  the  stu- 
dent. First  plot  the  measured  values,  then  find  the  ratio,  then  the 
equation,  and  proceed  to  correct  the  equation  pre- 
cisely as  before. 

2.   Make  a  similar  study  of  rolling   friction, 

with  the  aid  of  an  apparatus  like  the  one  in  the 

FIGURE  253  ,     -n-       c*~c\ 

cut,  Fig.  2o2. 

§216.  Thermometers. 

Fig.  253  shows  the  way  the  thermometer  tube  is  marked  off  and 
numbered  on  the  three  thermometers  most  used  by  civilized 
countries.  The  Fahrenheit,  or  common,  scale  is  on  the  left; 
the  Centigrade  scale,  used  everywhere  in  scientific  work,  is  on 
the  right;  and  the  Reaumur  (Ro'miir)  scale,  much  used  in  Ger- 
many, is  between  the  other  two. 

On  all  scales  the  fundamental  points  are  the  freezing-point 
where  the  top  of  the  mercury  column  stands  when  the  bulb  is 
in  water  containing  melting  ice,  and  the  boiling-point,  where  the 
top  of  the  mercury  column  stands  when  the  bulb  is  in  water 
commencing  to  boil.  For  any  temperature  the  reading  is  the 
number  which  belongs  to  the  mark  on  the  scale  at  the  top  of  the 
mercury  column. 


3(38 


RATIONAL    GRAMMAR    SCHOOL    ARITHMETIC 


REAUMUR 

-     10°  AsPhaU™  melts 


FIGURE  25-3 


1.  What  is  the   reading  for 
"Water  boils"  on  the  Fahren- 
heit scale?  on  the  Centigrade 
scale?  on  the  Reaumur  scale? 

2.  What  are  the  three  scale 
readings  for  ''Water  freezes"? 

3.  Into    how     many     equal 
parts  is  the  space  between  the 
boiling-point  and  the  freezing- 
point  divided  on  the  Reaumur 
scale?    on  the  Centigrade?  on 
the  Fahrenhei  t? 

4.  The  lengths  of  those  small 
spaces  are  marked  off  on  the 
tubes  above   the  boiling-point 
and  below  the   freezing-point 
and   numbered,    as   shown    in 
Fig.  253.        Notice    how    the 
numbers  run  on  each  of  the 
scales  between  the  freezing  and 
the   boiling   points;  below  the 
freezing    point.     How    should 
the  numbering  continue  above 
the  boiling  point? 

5.  What  do  we  call  a  change 
of  temperature  sufficient  to 
expand,  or  contract,  the  mer- 
cury column  an  amount  equal 
to  one  of  the  short  spaces  on 
the  Fahrenheit  scale?  on  the 
Centigrade  scale?  on  the  Reau- 
mur scale? 

These  are  written  1°F., 
i°G.,  and  1°R.,  and  are  read 
"1  degree  Fahrenheit,"  "1 
degree  Centigrade,"  and  "1 
degree  Heaumur. ' ' 


USES    OF    THE    EQUATION 


369 


6.  If  the  readings  above  CT°  are  written  +1°,  +2°,  +3°,  +5°,  etc., 
how  should  we  write  the  readings  below  0°? 

7.  What  do  the  -f  and  —  then  tell  us  about  the  readings? 

8.  Point  out  these  readings  in  the  figure:  +2°  F. ;  +36°C.; 
+29°R.;  -14°C.;  -20°R. ;  -38°F. ;  -10°F. 

9.  Give  the  readings  on  each  scale  for  the  mean  temperature 
at  the  Equator;  at  Eome;    London;    Chicago;   Edinburgh;   St. 
Petersburg;  at  the  Poles. 

10.  Which  degree  is  the  longest,  the  1°F.,  the  1°C.,  or  the 
1°R.?     Which  is  the  shortest? 

11.  How  many  Fahrenheit  degrees  are  equal  to  80°R.?     How 
many  Centigrade  degrees  equal  180°F.?     How  many  Reaumur 
degrees  equal  100 °C.? 

12.  On  a  sunny  day  find  how  many  Fahrenheit  degrees  warmer 
it  is  in  the  sun  than  in  the  shade?     Give  the  C  and  R  readings 
corresponding  to  the  two  F  readings  and  also  the  C  and  R  differ- 
ences. 


(o)      O 


00°' 


& 


§217.  Applied  Algebra  (Graduation  of  Thermometers) . 

The  problems  of  §213,  p.  360,  show  how  the 
use  of  the  equation  simplifies  some  kinds  of  arith- 
metic problems.  The  work  given  here  will  illus- 
trate how  some  problems  that  would  be  quite 
difficult  by  arithmetic  can  be  easily  handled  by 
means  of  the  equation. 

EXPERIMENTS.  —  To  convert  thermometer 
readings  from  one  scale  to  another: 

Three  thermometer  tubes,  exactly  alike,  were 
filled  with  mercury  to  the  same  height.  Their 
bulbs  were  immersed  in  a  vessel  containing  melt- 
ing ice.  Points  were  marked  on  the  tubes  at  the 
tops  of  the  mercury  columns;  beside  the  point 
on  the  first  tube  was  written  32,  and  beside  those 
on  the  second  and  third  tubes  0  was  written. 

All  three  of  the  bulbs  were  then  immersed  in 
water  commencing  to  boil,  and  the  tops  of  the 
mercury  columns  were  marked ;  beside  the  mark 
on  the  first  212  was  written;  beside  that  on  the 
second,  100;  and  beside  the  mark  on  the  third, 
80.  Let  AB  of  Fig.  254  denote  the  straight  line  passing  through 
the  tops  of  the  three  mercury  columns  when  the  bulbs  are  in 


F      C 

FIGURE  254 


370  RATIONAL   GRAMMAR   SCHOOL   ARITHMETIC 

melting  ice.  Let  CD  denote  the  line  passing  through  the  tops 
of  these  columns  when  the  bulbs  are  in  boiling  water. 

Suppose  the  space  lying  between  the  line  AH  and  the  line  CD 
on  the  first  tube  to  be  divided  into  180  equal  spaces  and  call  each 
of  these  small  spaces  one  degree  Fahrenheit  (written  1°F.). 
Denote  its  length  (in  in.  or  in  cm.)  by/. 

Suppose  this  same  space  on  the  second  tube  between  the  lines 
AB  and  CD  to  be  divided  into  100  equal  spaces,  and  call  one  of 
these  smaller  spaces  one  degree  Centigrade  (written  1°C.). 
Denote  its  length  (in  in.  or  in  cm.)  by  c. 

Divide  the  space  on  the  third  tube  between  the  lines  AB  and 
CD  into  80  equal  spaces,  and  call  one  of  these  spaces  one  degree 
Reaumur  (written  1°R.).  Denote  the  length  of  this  space 
(in  in.  or  in  cm.)  by  r. 

Notice  carefully  that  no  two  of  the  lengths  /,  c,  and  r  are 
equal.  Denote  the  length  of  the  space  from  AB  to  CD  by  S. 

1.  S  equals  how  many  times  f?c?  r? 

2.  Since  180  /  equals  S,  and  1006'  equals  S,  how  must  ISO/ 
compare  with  100  c?     (Answer  this  by  writing  an  equation.) 

This  problem  exemplifies  a  new  principle  of  much  importance 
in  using  equations  : 

PRINCIPLE  VI.  —  Numbers  that  are  equal  to  the  same  number 
are  equal  to  each  other. 

3.  Since  ISO/  equals  S,  and  80  r  also  equals  S,  how  must  ISO/ 
compare  with  80  r? 

4.  If  180  /=  100  c,  /equals  how  many  times  c? 

5.  If  ISO/  =  80  r,  /  equals  how  many  times  r? 

6.  From  these  two  equations,   giving  the  value  of  /,   write 
another  equation  by  the  aid  of  Principle  VI. 

7.  Explain  the  meaning  of  the  following  equations: 


Now  suppose  spaces  equal  to  /  marked  off  on  the  first  tube 
above  212  and  below  32,  spaces  equal  to  c  marked  off  on  the  second 
tube  above  100  and  below  0,  and  spaces  equal  to  r  marked  off  on 
the  third  tube  above  80  and  below  0.  The  first  tube  is  then  grad- 
uated to  the  Fahrenheit  scale,  the  second,  to  the  Centigrade  scale, 
and  the  third,  to  the  Reaumur  scale. 


USES    OF   THE    EQUATION  371 

§218.  Equivalent  Readings  on  the  Three  Thermometers. 

On  all  thermometers,  when  the  tops  of  the  mercury  columns 
are  above  0,  the  readings  are  called  positive,  and  are  marked  +; 
when  below  zero  they  are  called  negative,  and  are  marked  -. 

If  the  thermometers  are  exposed  to  the  same  temperature 
between  that  of  melting  ice  and  of  boiling  water,  the  tops  of  the 
mercury  columns  will  stand  in  a  straight  line,  as  XY.  Let  the 
mark  that  stands  by  this  line  at  the  top  of  the  column  on  the. first 
be  called  F\  on  the  second  top,  6';  and  on  the  third,  R. 

1.  Denoting  by  8  the  distance  from  the  line  AB  to  the  line 
XY  show  by  Fig.   254  that  (1)   S=f  (^-32);  (2)  S  =  Cv,  and 
(3)  S=Rr. 

2.  Show  by  Principle  VI  that  (4)  /  (F-  32)=  cC\  (5)  /  (F-  32) 
=  rR;  and  (6)  cC=r/i. 

3.  Show  from  equations  (1),  (2),   and  (3),  §217,  problem  7, 
that  (7)  c  =  f  /';  (8)  r  =  }/  and  (9)  r  =  f  c,  by  using  Principles  III 
and  IV. 

4.  In  equation  (4)  problem  2  substitute  c  =  f  /  from  (7)  problem 
3  and  by  Principles  IV  and  II  show  that  (I)   F=$C  +  3%. 

5.  In  a  similar  way  show  that  the  following  equations  are  true : 

(II)  ^=f#  +  32;  (V)    7?  =  |  (JF-32); 

(III)  C=t(F-M);  (VI)R  =  *C. 

(IV)  tf=iff; 

NOTE. — F,  C,  and  R  may  be  regarded  as  standing  respectively  for  any 
corresponding  readings  on  the  Fahrenheit,  Centigrade,  and  Reaumur 
thermometers. 

§219.  Problems. 

1.  Convert  the  following  into  their  F.  and  R.  equivalents  by 
the  aid  of  the  proper  equations  (I)  to  (VI). 

MELTING  TEMPERATURES  ON   CENTIGRADE  SCALE. 

Ice   .0.0°  Zinc 412.0° 

Benzol -}-  4.4°  Antimony 432.0" 

Tallow 43.0°  Silver  1000.0: 

Paraffin 46.0°  Copper 1100.0° 

Wax 62.0°  Gold 1200.0n 

Sulphur 115.0°  Cast  iron 1200.0° 

Tin  230.0°  Cast  steel 1375.0° 

Bismuth 250.0°  Wrought  iron 1600.0° 

Cadmium 320.0°  Platinum 1775.0° 

326  0^  Iridium 1950.0° 


372 


RATIONAL    GRAMMAR    SCHOOL    ARITHMETIC 


2.  Convert  these  Fahrenheit  boiling  temperatures  into  their 


C.  and  R.  equivalents: 

Ether 4-95° 

Carbon  Bisulphide  +  114.8° 

Sulphuric  Acid  ...  +14° 

Chloroform  .          .  +141.8° 


Alcohol +172.4' 

Water +212° 

Mercury +674.6C 

Zinc  .  .   + 1904° 


3.  Convert  any  observed  Fahrenheit  readings  to  their  Centi- 
grade and  Reaumur  equivalent  readings. 

4.  The  heat  of  the  body  of  a  healthy  man  is  37.2°C.     What 
are  the  Fahrenheit  and  Reaumur  equivalents? 

5.  Solid  carbonic  acid  dissolves  in  ether  and  the  solution  is  a 
liquid  of  -90°C.     What  is  the  Fahrenheit  equivalent? 

SOLUTION.— In  equation  (I)  if  C  =  -90°,  jtC  =  -162°.     Then  162°  meas- 
ured downward  and  32°  measured  upward,  or,  -162°  +  32'=  -130°. 

6.  For  every  vapor  and  gas  there  is  a  so-called  critical  temper- 
ature above  which  it  remains  in  the  gaseous  condition,  no  matter 
how  high  the  pressure  upon  it.     The  following  are  the  critical 
temperatures  of  some  vapors  and  gases:  Ether  vapor,  +196°C.; 
carbonic  acid,  +31  °C. ;  etheline,  +9°C. ;  oxygen,  — 118°C. ;  nitro- 
gen, -145°C.,  and  hydrogen, -174°C.    What  are  the  Fahrenheit 
equivalents? 

§290.  Laws  of  Thermometer  Represented  Graphically. 


When  it  is  necessary  to  change 
a  great  many  readings  from  one 
scale  to  another  it  is  convenient  to 
plot  the  equations.  To  illustrate 
this,  take  the  equation  F=$C  +  32. 
Draw  two  perpendicular  lines,  as 
OX  and  OY,  Fig.  255.  Let  all 
Centigrade  readings  be  measured  off 
horizontally  and  the  corresponding 
Fahrenheit  readings  vertically  to 
any  convenient  scales. 
—  The  equation  shows  that  if 

50  CENTIGRADE  READINGS  <7=  0°,    F=  32°  J  if  C=  20°,  F=  68°  ; 

and  so  on.     In  this  way  we  fill  out 
FIGURE  255  the  following  table : 


71 


SHORTENING   AND   CHECKING   CALCULATIONS  373 

C.  F.                       C.  F. 

0°  32°  100°  212° 

20°  68°  200°  392° 

40°  104°  -20°  -4° 

50°  122°  -40°  -40° 

75°  167°  -80°  -112° 

NOTE. — The  numbers  preceded  by  the  sign  —  are  readings  below 
zero,  and  such  numbers  for  C.  must  be  measured  off  from  0  toward  the 
left,  and  for  F.  they  must  be  measured  off  downward. 

Study  the  points  on  Fig.  255  and  note  whether  they  seem  to  lie  on  a 
straight  line.  With  a  ruler  draw  a  straight  line  through  these  points. 

For  minus  ( — )  values  of  C,  as  for  C=  — 20,  proceed  thus:  F=%  X 
(—20)  +  32  =  9  X  (—¥)+-  32  =  9  X  (—  4)  -(-  32  =  —36  -f  32.  But  36°  meas- 
ured downward  and  then  32°  measured  upward  is  the  same  as  4° 
measured  downward,  or  F=  —4°,  if  C  =  — 20°;  and  similarly  for  other 
minus  (— )  values  of  C. 

1.  Measure   off  the  value   C  =  60°,    then   measure  vertically 
upward  to  the  line  drawn  through  the  points,  thus  obtaining  the 
corresponding  Fahrenheit  reading.     What  is  F.  for  C.  =  60°? 

2.  Similarly,  find  from  the  drawing  the  values  of   F.  corre- 
sponding to  these  values  of  C.:    120°;  160°;    180°;  220°;  -60°; 
-100°;  110°. 

3.  Make  a  similar  table  and  plot  of  one  or  more  of  the  other 
five  equations  of  problem  5,  p.  371. 


METHODS   OF   SHORTENING   AND   CHECKING 

CALCULATIONS 
§221.  Illustrations, 

1.  Additions,  subtractions,  multiplications,  and  divisions  are 
conveniently  checked  by  casting  out  the  9's,  as  explained  on  pp. 
42,  66,  and  87. 

2.  To  check  against  gross  errors  (blunders),  first  think  through 
the  problem,   making  rough  mental   calculations  with  numbers 
that  are  approximately  correct,  and  decide  about  what  the  result 
must  be. 

3.  Check   by  performing   reverse   operations;    that  is,  check 
addition  by  adding  columns  in  reverse  order;  check  subtraction 
by  adding  the  subtrahend  to  the  remainder;   check  division  or 
square  and  cube  roots  by  multiplication,  and  so  forth. 


RATIONAL    GRAMMAR    SCHOOL   ARITHMETIC 

The  second  rule  may  be  illustrated  by  a  few  problems  : 
1.  Find  the  value  of  15f  A.  of  land  at 


Think  thus:  16  A.  @  $90  would  be  worth  $1440;  at  $85,  16  A.  would  be 
worth  $80  less,  or  $1360.  At  $87£,  15|  A.  of  land  is  worth  i  of  $85  less 
than  $1440—i  of  $80  (=$1400),  or  about  $1383.  The  exact  computation 
gives  $1382.50. 

Or  thus:  87£  =  I  of  100.      Therefore  15$  X  87|  =  1580  X  I  =  7  X  197.5 
=  $1382.5.     Ans.  $1382.5. 

2.  On  June  13,  1903,  wheat  is  quoted  in  Chicago  at  76f^  per 
bushel.     Find  the  cost  of  150  bu.  at  this  price. 

3.  250  shares  of  Illinois  Central  R.  R.  stock  sold  at  $135^  a 
share.     For  how  much  did  they  sell? 

4.  The  diameter  of  a  circular  rod  is  If".     How  many  inches 
in  the  circumference  of  a  right  section  of  the  rod?     How  many 
square  inches  are  there  in  the  area  of  a  right  section  of  the  rod? 
(For  an  approximation  use  TT  =  3|.) 

5.  The  outside  diameter  of  a  circular  hollow  iron  tube  is  2£" 
and  the  inside  diameter  is  2".     How  many  cubic  inches  are  there 
in  a  12'  length  of  the  tube? 

6.  A  distance  was  measured  with  a  chain  98.75  ft.  long  and 
was  found  to  contain  38.75  lengths  of  the  chain.     The  chain  was 
supposed  to  be  100'  long.      How  great   an   error   was   made   in 
measuring  the  line  by  using  the  supposed  length? 

§222.  Shortening  and  Checking  Addition. 

1.  The  noon  temperatures  on  7  successive  days  were  GG°,  54°, 
44°,  G2°,  66°,  79°,  88°.     Find  the  average  for  the  week. 

66  MENTAL  WORK.     70  +  50  +  44  =  164;   164  -f  60  = 

54  224;    224+2  =  226;    226+70  =  296;     296—  4=292; 

44     164  292  -f  80  =  372;  372  —  1  =  371  ;  371  +90  =  461  ;  461  — 

226    62  2  =  459. 

66    292  This  is  called  two-column  addition.     A  little  prac- 

39  1     79  tice  will  make  this  method  useful  for  1,  2,  or  3  columns 

g$  of  figures.     It  may  be  used  to  advantage  to  check  addi- 

—  —  -  tions  made  in  the  ordinary  way. 

7_)459  Another  method  in  use  by  expert  accountants  is  to 

65  6°     Ans      group  the  figures  into  sums  of  10,  20,  30,  and  so  on. 
Thus,  in  the  given  problem,  8  +  2,  6  +  4,  6  +  4,  and  9 
make  39.     Then  7  +  3,  8  +  6  +  6,  6  +  4,  5,  are  45.     Sum,  459. 

2.  Write  a  few  two  and  three  column  aclditi  on  .  problems  and 
practice  these  methods  until  you  can  use  them  rapidly. 


SHORTENING    AND    CHECKING    CALCULATIONS  375 

§223.  Making- up  Method  of  Subtraction. 

1.  A  paying-teller  in  a  bank  had  $5485  in  his  cash  drawer  in 
the  morning,  and  during  the  day  he  paid  out  the  following 
amounts:  $37.50;  $5.60;  $165.75;  $10.25;  $3.50;  $2.88;  $1.76; 
$65.17;  $968.23;  $3.67.  How  much  money  remained  in  the 
drawer? 

CONVENIENT  FORM.  MENTAL   WORK.— Add  the   first    column, 

Total,  $5485.00  thinking  thus:  10,  28,  31,  41.     41  and  9  make 

the  next  larger  number  than  41  ending  in  0 
3^.50  (the  first  figure  in  the  total.)     Write  the  9  in 

165. 75  the  result  and  add  the  5  into  the  second  col- 

umn. Then  11,21,34,48.  48  -f  2  =  50,  the 
next  number  larger  than  48  which  ends  in  0 
(the  second  figure  of  the  total).  Write  2  in 
the  result  and  add  5  into  the  third  column. 
Then,  21,  31,  39.  39  +  6  =  45.  Write  6  and 
968.23  ad(j  4  into  fourth  column.  Then  10,  17,  26.  26  + 

3-67  2  =  28.     Write  2  and  add  2  to  next  column. 


$4226.29,  balance        Then,  12.     12  +  2  =  14,  and  finally,  1  +  4  =  5. 

Write  the  4. 

The  advantage  of  this  method  is  that  it  foots  all  the  numbers  and 
subtracts  their  sum,  at  once,  as  the  numbers  stand  in  the  account  book, 
from  the  total,  giving  the  balance  directly. 

2.  A  bank  customer's  deposit  at  the  beginning  of  the  month 
was  $398.75,  During  the  month  he  drew  out  the  following 
amounts:  $16.75;  $1.75;  $5.25;  $12.87;  $128.32;  $40.45; 
$2.18;  $9.16;  $1.57;  $11.38;  $12.62.  Find  the  customer's 
balance  at  the  end  of  the  month. 


§224.   Shortened  Multiplication. 

1.  Multiply  73  by  67. 

67  =  70^3  I  73  X  67  =  (70 +  3)  (70  -3)  =  702  +  W  X  3  —  £  X  W  —  32  = 

702  —  32  =  4900  —  9  =  4891. 
Algebraic  form :  (a  -f-  6)  (a  —  6)  =  a2  —  62. 

This  applies  to  finding  the  product  of  any  two  numbers  the  sum  of 
whose  units  is  10  and  whose  tens  digits  differ  by  1. 

EULE. — Find  the  difference  between  the  square  of  the  tens  and 
the  square  of  the  units  in  the  larger  number. 


376  RATIONAL   GRAMMAR    SCHOOL   ARITHMETIC 

2.  Find  these  products  by  the  rule: 

(1)  34  x  26.  (4)  58  x  42.  (7)  71  x  69. 

(2)  22  x  18.  (5)   65  x  55.  (8)  99  x  81. 

(3)  43  x  37.  (6)  98  x  82.  (9)  95  x  85. 

3.  Find  the  square  of  38. 

38s  =  (30  +  8)2  =  302  +  2  X  8  X  30  +  82  =  1396 
Algebraic  form :  (a  -+-  b)2  =  a2  +  2ab  +  Z>2. 

Show  the  correctness  of  the  following  rule : 

EULE. — Square  the  tens,  double  the  product  of  the  tens  by  the 
units  and  square  the  units,  then  add  the  three  results.  The  sum 
is  the  square  of  the  number. 

4.  Square  these  numbers  by  the  rule : 

(1)  36.  (3)  64.  (5)  19.  (7)  125. 

(2)  47.  (4)  58.  (6)  95.  (8)  148. 

NOTE.— Call  the  12  in  (7)  12  tens;  also  call  the  14  in  (8)  14  tens. 

36  might  be  written  40  —  4.  Then  362  =  (40  —  4)2,  (40  —  4)2  =  402  — 
2  X  4  X  40  -(-  42.  Algebraic  form :  (a  —  6)2  =  a2  —  2  ab  +  b'2.  Make  a  rule 
for  squaring  36  in  this  form. 

When  long  decimals  are  to  be  multiplied  and  the  product  is 
required  to  only  a  few  decimal  places,  contracted  multiplication  is 
of  great  advantage.  The  next  problem  illustrates  this  sort  of 
multiplication. 

5.  The  radius  of  a  circle  is  238.36  ft.     What  is  the  length  of 
the  circumference  to  the  second  decimal  place,  or  to  the  nearest 
.01  foot? 

NOTE.— The  length  of  the  diameter  is  476.72  feet. 

COMMON  FORM  SHORTENED  FORM 

476.72  476.72 

3.1416  6141.3  digits  reversed 


28 
47 

1906 

4767 

143016 


1497.66 


6032  1430.16 

672  47.67 

88  19.07 

2  48 

29 


3552  1497.67 


SHORTENING    AND    CHECKING   CALCULATIONS 


377 


EXPLANATION. — In  the  shortened  form  the  digits  of  the  multiplier  are 
written  in  reverse  order,  and  the  units  digit  is  always  written  under  that 
decimal  place  in  the  multiplicand  which  is  to  be  the  last  one  retained  in 
the  product. 

Multiply  by  units  digit  first,  then  by  tens,  and  so  on;  in  each  case 
begin  the  multiplication  by  any  digit  with  the  digit  just  above  it  in 
the  multiplicand.  Begin  the  writing  of  each  partial  product  in  the  same 
vertical  line  on  the  right. 

NOTE. — It  is  necessary  on  beginning  to  multiply  by  any  digit  to 
glance  at  the  product  by  the  preceding  digit  of  the  multiplicand  to  see 
how  many  units  are  to  be  added  into  the  product  by  the  digit  just  above. 
Thus,  the  multiplication  by  4  would  begin  with  6,  but  4  times  the  pre- 
ceding digit  (7)  is  28,  and  this  being  nearly  3,  the  product  4x6  would 
be  increased  by  3,  giving  27. 

Expert  computers  use  the  shortened  form  altogether. 


3.  Eind  the  following  products  to  the  second  decimal  place  by 
the  method  of  shortened  multiplication: 


(1)  30.428  x  3.1416. 

(2)  186.086  x  108.336. 


(3)  7.8843  x  1.0863. 

(4)  168.7431  x  28.329. 


4.  Find  these  products  to  .001  by  shortened  multiplication: 


(1)  36.1872  x  6.8734. 

(2)  128.63  x  3.8629. 


(3)  629.3865  x  3.1416. 

(4)  1284.683x3.1416. 


$25.  Shortened  Division. 

1.  Divide  648.7863  by  68.372  to  the  nearest  .01. 


CONVENIENT  FORM 

9.49—    Quotient 

68.37)648.7863 
61533 

3345 


610 
6  15 


EXPLANATION.— Find  the  units  digit  of 
the  quotient  in  the  usual  way.  Then  cut  off 
one  digit  from  the  right  of  the  divisor  and 
find  the  next  digit  of  the  quotient,  then  cut 
off  another  digit  from  the  divisor,  etc.  A  dot 
is  sometimes  placed  over  each  digit  in  the 
divisor  as  it  is  set  aside. 


2.  Find  the  following  quotients  to  two  decimal  places: 

(1)  1786.786-3.1416. 

(2)  632.068-8.6249. 

(3)  1206.3862-28.3762. 

(4)  865.28476  +  361.2946. 


378 


'    RATIONAL    GRAMMAR    SCHOOL    ARITHMETIC 


§226.  Shortened  Square  Boot. 

Square  root  may  be  found  by  subtraction,  after  noticing  that 
1  =  12;  l  +  3  =  23;  l  +  3  +  5  =  32;  H-3  +  5  +  7  =  4a;  and  so  on. 

It  is  seen  that  the  square  root  of  the  sum  of  the  odd  numbers 
in  order  from  1  upward  is  equal  to  the  number  of  odd  numbers 
added. 

The  method*  of  using  this  can  be  best  understood  from  an 
example. 

1.   Extract  the  square  root  of  104976. 


104976  ( 324 
1 

9 
3 


01 


641 


149 
61 

88 
63 


2576 
641 


3  subtractions 


2  subtractions 


1935 
643 

1292 
645 

647       4  subtractions 
324  =  square  root. 


EXPLANATION.— Place  a  dot  over  each 
alternate  digit  beginning  on  the  right, 
thus  separating  the  number  into  groups 
of  2  digits  each.  From  the  first  group  on 
the  left  subtract  1,  then  3,  then  5,  and  so 
on  as  shown,  until  the  remainder  is  less 
than  the  next  odd  number.  The  number 
of  subtractions  is  the  first  root  digit.  In 
the  present  case  it  is  3. 

Bring  down  the  next  two-digit  group. 
Double  the  root  digit  found  and  annex 
1  to  it,  and  subtract,  then  replace  the  1 
by  3  and  subtract,  etc.  The  number  of  sub- 
tractions indicates  the  next  root  digit. 

Study  the  remaining  steps  and  learn 
how  to  proceed  further.  This  method 
gives  the  exact  square  root  and  may  be 
used  to  check  results  found  in  the  ordinary 
way.  Some  computing  machines  are  based 
on  this  method  of  obtaining  the  square 
root. 


2.  Find  the  square  roots,  by  subtraction,  of  the  following: 

(1)  1156.  (3)  174.24.  (5)    54,756. 

(2)  44,944.  (4)   49,284.  (6)   289444. 


SYNOPSIS  OF  DEFINITIONS. 

4.  A  one-brick  wall  is  a  wall  one  brick  thick,  the  bricks  lying  on  the 

largest  surfaces,  the  sides  being  exposed. 
10.  The  average  of  two  or  more  numbers  is  their  sum  divided  by  the 

number  of  them. 
12.  A  board  foot  is  a  board  one  foot  long,  one  foot  wide,  and  not  more 

than  one  inch  thick. 

14.  Cash  rent  of  farming  land  is  a  stated  amount  of  cash  per  acre. 
Grain  rent  of  farming  land  is  a  stated  part  of  all  the  crop. 

The  tenant  farmer  is  one  who  raises  his  crop  on  another  man's  farm. 

15.  What  per  cent  means  "how  many  in  a  hundred"  or  "how  many 

hundredths."     (Cf.  p.  130.) 
17.  Normal  means  average  here. 

The  mean  of  two  numbers  is  half  their  sum. 

22.  To  average  means  to  find  the  average. 

23.  The  range  of  temperature  is  the  difference  between  the  highest  and 

the  lowest  temperatures. 
26.  The  reading    of  a  barometer  is  the  difference  between  the  lengths 

of  the  mercury  columns. 
28.  The  range  of  the  barometer  is  the  difference  between  the  greatest  and 

the  least  readings. 

31.  The  digits  (or  figures)  are  the  ten  characters  0,  1,  2,  3,  4,  5,  6,  7,  8,  9. 
The  name  value   of   a  digit  is  the  value  depending  only  upon   the 

name  of  the  digit. 
ThepZac-e  value  of  a  digit  is  the  value  depending  only  upon  the  place 

of  the  digit. 
34.  The  index   notation  is  such  a  form  as  675xl012,  meaning  675,000,- 

000,000,000. 

36.  Addition  is  combining  numbers  into  a  single  number. 
The  sum  or  amount  is  the  result  of  the  addition. 
The  addends  are  the  numbers  to  be  combined  or  added. 
47.  Subtraction  means  either  of  two  things : 

(1)  The  way  of  finding   the   difference,  or  remainder,  of  two 
numbers. 

(2)  The  way  of  finding  either  one  of  two  addends  when  their 
sum  and  the  other  addend  are  known. 

With  the  first  meaning,  the  number  from  which  we  subtract 
is  the  minuend.  The  number  to  be  subtracted  is  the  subtrahend. 
The  result  is  called  the  difference  or  remainder. 

With  the  second  meaning,  the  known  sum  is  the  minuend, 
379 


380  RATIONAL    GRAMMAR    SCHOOL    ARITHMETIC 

PAGE 

The  known  addend  is  the  subtrahend,  and  the  unknown  addend  is 
the  difference  or  remainder. 

57.  Literal  numbers  are  those  that  are  denoted  by  letters, 

58.  Multiplication  of  whole  numbers  is  a  short  way  of  finding  the  sum 

of  equal  addends  when  the  number  of  addends  and  one  of  them 
are  given. 

The  given  addend  is  the  multiplicand. 

The  number  of  equal  addends  is  the  multiplier. 

The  result  or  sum  is  the  product. 

When  a  problem  is  expressed  in  an  equation,  the  equation  is 
called  the  statement  of  the  problem. 

The  number  which  should  stand  in  place  of  x  in  an  equation 
is  called  the  value  of  x. 

63.  A  factor  of  a  given  whole  number  is  one  of  two  or  more  whole  num- 

bers, which,  multiplied  together,  produce  the  given  number. 
(Cf.  p.  148.) 

64.  A  number  which  has  factors  other  than  itself  and  1  is  called  a  com- 

posite number. 
A  number  which  has  no  factors  other  than  itself  and  1  is  a  prime 

number. 
69.  One  inch  of  rainfall  means  one  cubic  inch  of  water  for  each  square 

inch  of  horizontal  exposed  surface. 
The  number  of  cubic  units  (cu.  in.,  cu.  ft.,  cu.  yd.,  etc.)  a  vessel 

holds,  when  full,  is  called  its  capacity. 

73.  Division  is  a  short  way  of  finding: 

(1)  One  of  a  given  number  of  equal  parts  of  a  number. 

(2)  How  many  equal  parts  of  given  size  there  are  in  a  given 
number. 

When  the  term  "division"  has  the  meaning  of  (2),  the  process 
to  which  it  applies  may  be  called  measurement. 

Division  of  whole  numbers  is  a  short  way  of  subtracting  one 
number  from  another  a  certain  number  of  times  in  succession. 
(Cf.  p.  74.) 

74.  Division  is  a  way  of  finding  one  of  two  numbers  when  their  product 

and  the  other  number  are  given 

The  product  is  called  the  dividend. 
The  given  number  is  the  divisor. 
The  required  number  is  the  quotient. 

We   may  say  also  that    the  dividend  is  the  number  to  be 
divided,  the  divisor  is  the  number  by  which  the  dividend  is  meas- 
ured or  divided,  and  the  quotient  is  the  measure.     (Cf.  p.  73.) 
89.  The  point  on  which  a  lever  rests  is  called  the  fulcrum. 
103.  Such  an  expression  as  y=10  is  called  an  equation. 

The  number  on  the  left  of  the  sign  of  equality  is  called  the  first 


SYNOPSIS    OP   DEFINITIONS  381 

PAGE 

member,  or  the  left  side,  of  the  equation.  The  number  on  the 
right  is  called  the  second  member,  or  the  right  side,  of  the  equa- 
tion. 

104.  The  pencil  point  of  the  compasses  is  called  the  pencil-foot  or  the  pen- 
foot.     The  other  point  is  called  the  pin-foot. 

A  circle  is  a  curve  such  as  is  drawn  with  compasses. 

An  arc  of  a  circle  is  a  part  of  a  circle. 

The  center  of  a  circle  is  the  point  where  the  pin-foot  of  the  compasses 
was  placed  to  draw  the  circle. 

A  diameter  of  a  circle  is  a  straight  line  joining  two  points  of  the 
circle  and  passing  through  the  center. 

A  radius  of  a  circle  is  a  straight  line  joining  its  center  and  a  point 
of  the  curve. 

Circles  whose  centers  are  at  the  same  point  are  called  concentric 
circles. 

107.  Dividing  a  line  into  two  equal  parts  is  called  bisecting  the  line. 

108.  An  equilateral  triangle  is  an  equal-sided  triangle. 
An  isosceles  triangle  has  at  least  two  sides  equal. 

That  side  of  an  isosceles  triangle  not  equal  to  either  of  the  other  two 
sides,  is  called  the  base  of  the  isosceles  triangle. 

109.  A  scalene  triangle  has  no  two  sides  equal. 

110.  A  regular  hexagon  is  a  regular  six-sided  figure. 

111.  The  sum  of  all  the  bounding  lines  of  a  figure  is  called  the  perimeter 

of  the  figure. 

Money  is  the  common  measure  of  the  value  of  all  articles  that  are 
bought  and  sold. 

113.  One   of  a  whole   number  of  equal  parts  of  a  given  magnitude  is 

called  a  fractional  unit.     (Cf.  p.  144.) 

114.  A  square  unit  is  a  square  one  unit  long  and  one  unit  wide. 

115.  A  representation  of  an  object,  showing  the  various  parts  as  folded 

back  and  spread  out  on  a  flat  surface,  is  called  a  development. 
A  quadrilateral  is  a  four-sided  figure. 

A  parallelogram  is  a  quadrilateral  whose  opposite  sides  are  parallel. 
A  rectangle  is  a  parallelogram  whose  angles  are  (equal)  right  angles. 
A  rhombus  is  a  parallelogram  whose  sides  are  equal. 
A  square  is  both  a  rectangle  and  a  rhombus. . 

116.  The  altitude  of  a  parallelogram  is  the  distance  square  across. 

117.  The  altitude  of  a  triangle  is  the  shortest  distance  to  the  base  from 

the  opposite  corner.  With  all  except  isosceles  triangles  any  side 
may  be  regarded  as  the  base. 

The  altitude  of  a  trapezoid  is  the  distance  square  across  between  two 
parallel  sides  called  the  bases  of  the  trapezoid. 

118.  The  lengths  of  the  bases  and  of  the  altitude  of  a  trapezoid  are  called, 

its  dimensions. 


o8->  RATIONAL   GRAMMAR   SCHOOL   ARITHMETIC 

PAGE 

124.  The  mean  solar  day  is  the  average  time  interval  during  which  the 

rotation  of  the  earth  carries  the  meridian  of  a  place  eastward 
from  the  sun  back  around  to  the  sun  again.  It  is  the  average 
length  of  the  interval  from  noon  to  the  next  noon.  (Of.  p. 
229.) 

125.  A  section  of  land  is  a  square  tract  containing  640  acres. 

126.  A  township  is  a  tract  of  land  six  miles  square. 

130.  Per  cent  means  hundredth  or  hundredths.     (Of.  p.  15.) 

131.  Interest  is  money  to  be  paid  for  the  use  of  money.     (Cf.  p.  258.) 

138.  The  average  rate  of  running  is  the  distance  divided  by  the  time. 

139.  A  diagonal  of  a  figure  is  a  straight  line,  not  a  side,  joining  two 

corners. 

1 12.  The  ratio  of  one  number,  (or  quantity),  to  a  second  number,  (or 
quantity),  is  the  quotient  of  the  first  number,  (or  quantity), 
divided  by  the  second. 

The  ratio  of  one  magnitude  to  a  second  magnitude  is  called  also  the 
measure  of  the  first  by  the  second. 

The  result  of  measuring  one  number  by  another  is  called  the  numer- 
ical measure  of  the  first  by  the  second. 

An  equation  of  ratios  is  called  a  proportion. 

144.  The  fractional  unit  of  a  fraction  is  one  of  the  equal  parts  expressed 

by  the  fraction.     (Cf.  p.  113.) 

145.  The  number  above  the  fraction  line  is  called  the  numerator  (mean- 

ing numberer}. 
The  number    below    the  fraction   line    is  called    tne  denominator 

(meaning  namer). 
The  numerator  and  the  denominator  are  together  called  the  terms  of 

a  fraction. 

147.  A  fraction  is  said  to  be  in  its  lowest  terms  when  the  numerator  and 

the  denominator  are  the  smallest  possible  whole  numbers  without 
changing  the  value  of  the  fraction. 

The  greatest   common  divisor   (G.  C.  D.)  of  two  numbers  is  their 
greatest  exact  common  divisor. 

148.  Two  numbers  that  have  no  common  factor,  except  1,  are  said  to  be 

prime  to  each  other. 

A  factor  of  a  number  is  an  exact  divisor  of  the  number,  or  a  divisor 
that  is  contained  without  a  remainder.     (Cf.  p.  63.) 

149.  A  number  that  has  no  factors  except  itself  and  1  is  called  a  prime 

number. 
A  number  that  has  factors  beside  itself  and  1  is  called  a  composite 

number. 
Any  number  that  can  be  exactly  divided  by  the  number  2  is  called 

an    even    number.      All    other  whole   numbers    are  called    odd 

numbers. 


SYNOPSIS    OF    DEFINITION'S  383 

PAGE 

156.  The  average  growth  from  both  lower  and  middle  branches  of  a  tree 

is  called  the  lateral  growth.     The  average  from  the  top  branches, 
is  called  the  terminal  growth. 

157.  A  denominator  which  is  common  to  two  or  more  fractions  is  called  a 

common  denominator. 

When  a  common  denominator  is  the  least  number  that  can  be  found 
which  may  be  used  as  a  common  denominator  of  the  fractions,  it 
is  called  the  least  common  denominator  (L.  C.  D.). 

158.  A  number  that  can  be  exactly  divided  by  another  number  is  called  a 

multiple  of  the  latter  number. 

A  number  that  can  be  exactly  divided  by  two  or  more  numbers  is 
called  a  common  multiple  of  those  numbers. 

159.  The  least  common  multiple  (L.  C.  M.)  of  two  or  more  numbers  is  the 

least  whole  number  that  is  exactly  divisible  by  each  of  the  num- 
bers. 

161.  A  proper  fraction  is  a  fraction  whose   numerator  is  less  than  its 

denominator. 
An  improper  fraction  is  a  fraction  whose  numerator  is  equal  to,  or 

greater  than,  its  denominator. 

A  mixed  number  is  a  number  that  is  composed  of  an  integer  and  a 
fraction. 

170.  To  multiply  a  whole  number  by  &  fraction  means  to  divide  the  multi- 
plicand into  as  many  equal  parts  as  there  are  units  in  the 
denominator,  and  to  take  as  many  of  these  equal  parts  as  there  are 
units  in  the  numerator,  of  the  multiplier. 

180.  Dividing  1  by  any  number,  whole  or  fractional,  is  called  inverting 

the  number. 
The  reciprocal  of  any  number  is  the  number  inverted. 

182.  Fractions  containing  fractions  in  one  or  both  terms  are  called  com- 
plex fractions. 

The  outside  terms  of  a  complex  fraction  (such  as-?-)  are  called  the 

15T 

extremes  and  the  inside  terms  are  called  the  means. 
186.  A  straight  line  connecting  the  mid-point  of  a  side  of  a  triangle  with 

the  opposite  corner  is  called  a  median  of  the  triangle. 
The  vertex  of  an  angle  is  the  corner.     (The  plural  of  vertex  is  vertices 
(ver'-tises.)    (Cf.  pp.  278,  292.) 

188.  To  trisect  a  magnitude  is  to  divide  it  into  three  equal  parts. 

189.  A  right-angled  triangle  is  called  a  right  triangle. 

190.  Parallel  lines  are  lines  running  in  the  same  direction. 

192.  A  square  is  inscribed  in  a  circle  if  the  vertices  of  the  square  are  all 

on  the  curve. 
196.  The   first,  second,  third,  and  fourth  numbers  of  a  proportion  are 

called  the  first ,  second,  third,  and  fourth  terms  of  the  proportion. 


384  RATIONAL    GRAMMAR   SCHOOL    ARITHMETIC 

PAGE 

The  first  and  the  fourth  terms  of  a  proportion  are  called  the  extremes 

and  the  second  and  the  third  terms,  the  means. 
The  first  two  terms  of  a  proportion  are  together  called  the  first 

couplet ;  and  the  third  and  the  fourth  terms,  the  second  couplet. 

200.  A  dot,  called  the  decimal  point,  or  point,  is  used  to  show  the  units' 

digit.     The  point  always  stands  just  to  the  right  of  the  units' 
digit  or  place. 

201.  The  unit  of  the  1st  place,  or  digit,  to  the  right  of  the  decimal  point 

is  called  the  tenth;  of  the  2d  place,  or  digit,  the  hundredth;  of  the 
3d,  the  thousandth;  of  the  4th,  the  ten-thousandth;  and  so  on. 

202.  A  decimal  fraction,  or  decimal,  is  a  fraction  whose  denominator  is 

10,  100,  1000,  or  some  power  of  ten,  in  which  the  denominator  is 
not  written  but  is  indicated  by  the  position  of  the  decimal  point. 
A  power  of  10  here  means  a  number  obtained  by  using  10  as  a  factor 
any  whole  number  of  times. 

203.  A  pure  decimal  is  a  decimal  whose  value  is  less  than  one. 

A  mixed  decimal  is  a  decimal  whose  value  is  greater  than  one. 
Numbers   expressed  in    both  decimals  and  common  fractions  are 

called  complex  decimals. 

A  simple  decimal  is  expressed  without  the  use  of  common  fractions. 

Finding  the  sum  of  decimal  numbers  is  called  addition  of  decimals. 

205.  Finding  the  difference  of  decimal  numbers  is  called  subtraction  of 

decimals. 
208.  By  the  number  of  decimal  places  of  a  number  is  meant  the  number 

of  digits  (zero  included)  on  the  right  of  the  decimal  point. 
214.  The  distance  round  a  circle  is  sometimes  called  the  circumference 

of  the  circle. 
217.  The  specific  gravity  of  any  solid  or  liquid  substance  is  the  ratio  of 

its  weight  to  the  weight  of  an  equal  bulk  of  water. 

219.  Decimals  that  do  not  terminate  are  called  non-terminating  decimals. 
Non-terminating  decimals   that   repeat  a  digit  or  group  of  digits 
indefinitely  are  called  repetends,  or  circulating  decimals,  or  cir- 
culates. 

221.  A  right  section  is  a  section  made  by  cutting  squarely  across. 
223.  A  safety  bicycle  is  said  to  be  geared  to  72",  84",  and  so  on,  when  one 
turn  of  the  cranks  would  carry  it  just  as  far  forward  as  would  one 
turn  of  a  wheel  having  a  diameter  of  72",  84",  and  so  on. 
225.  A  denominate  number  is  a  number  whose  unit  is  concrete. 
A  concrete  unit  is  a  unit  having  a  specific  name. 
A  compound  denominate  number  is  a  number  expressed  in  two  or 

more  unite  of  one  kind. 
227.  A  perch  of  stone    is  a  square-cornered  mass,   l'Xl>2'Xl6^' 

cubic  feet. 
A  cord  of  firewood  is  a  straight  pile,  4'X4'X8'=128  cubic  feet. 


SYNOPSIS   OF   DEFINITIONS  385 

PAGE 

A  cord  foot  is  a  straight  pile  of  wood,  4'x4'Xl'=16  cubic  feet. 

229.  A  leap  year  is  a  year  of  366  days. 

A  common  year  is  a  year  of  365  days. 
232.  Reduction  from  higher  to  lower  denoniinations  is  called  reduction 

descending. 
Reduction  from  lower  to  higher  denominations  is  called  reduction 

ascending. 

236.  A  meter  is  approximately  one  ten-millionth  of  the  length  of  the  part 
of  a  meridian  of  the  earth,  that  reaches  from  the  equator  to  the 
pole,  called  a  quadrant  of  the  earth's  meridian. 
238.  An  are  is  one  square  dekameter. 
A  liter  is  one  cubic  decimeter. 
A  gram  is  the  weight  of  one  cubic  centimeter  of  distilled  water  at 

the  temperature  of  its  greatest  density  (39.1°  Fahrenheit). 
242.  The  number  written  before  the  sign  "%"  is  called  the  rate  per  cent. 
The  number,  together  with  the  sign,   "%,"  is  called  the  rate. 
(Cf.  p.  258.) 
245.  The  result  of  finding  a  given  per  cent  of  any  amount,  or  number,  is 

called  the  percentage. 
The  amount,  or  number,  on  which  the  percentage  is  computed  is 

called  the  base. 
249.  Elevation  means  height  above  mean  sea  level. 

254.  Commission  is  a  sum  of  money  paid  by  a  person  or  firm,  called  the 

principal,  to  an  agent  for  the  transaction  of  business.  It  is 
usually  reckoned  as  some  per  cent  of  the  amount  of  money 
received  or  expended  for  the  principal. 

A  shipment  of  goods  sent  to  an  agent  to  be  sold  is  called  a  consign- 
ment. 

255.  The  net  proceeds  of  a  sale  means  the  amount  left  after  deducting  the 

commission  and  other  expenses. 

A  commercial  (or  a  trade)  discount  is  a  certain  rate  per  cent  of 
reduction  from  the  listed  prices  of  articles.  The  discount  is 
usually  allowed  for  cash  payments  'or  for  payment  within  a 
specified  time. 

258.  Interest  is  money  charged  for  the  use  of  money.  It  is  reckoned  at  a 
certain  rate  per  cent  of  the  sum  borrowed  for  each  year  it  is  bor- 
rowed. (Cf.  p.  258.) 

When  money  earns  3,  6,  7,  or  10  cents  on  the  dollar  annually  (each 
year)  the  rate  is  said  to  be  3 %,  6%,  7%,  or  10%  per  annum  (by  the 
year)  and  the  rate  per  cent  is  said  to  be  3,  6,  7,  or  10.  (cf.  p.  242. ) 

The  sum  of  money  on  which  interest  is  computed  is  called  the 
principal. 

The  principal  plus  the  interest  is  called  the  amount. 
265.  A  promissory  note  is  a  written  promise,  made  by  one    person    or 


386  RATIONAL    GRAMMAR   SCHOOL   ARITHMETIC 

PAGE 

party,  called  the  maker,  to  pay  another  person  or  party,  called 

the  payee,  a  specified  sum  of  money  at  a  stated  time. 
The  sum  of  money  for  which  the  note  is  drawn  is  called  the  face 

value,  or  the  face,  of  the  note. 
The  date  on  which  the  note  falls  due  is  called  the  date  of  maturity, 

and  the  time  to  run  from  any  given  date  is  the  time  yet  to  elapse 

before  the  note  falls  due. 

266.  Discount  is  a  deduction  from  the  amount  due  on  a  note  at  the  date 

of  maturity. 
The  sum  of  money  which,  at  the  specified  rate  and  in  the  time  the 

note  is  to  run  before  falling  due,  will,  with  interest,  amount  to 

the  value  of  the  note  when  due,  is  called  the  present  worth  of  the 

note. 
The  difference  between  the  value  of  the  note  when  due,  and  the 

present  worth  is  called  the  true  discount. 
The  bank  discount  of  a  note  is  the  interest  upon  the  value  of  the  note 

when  due,  from  the  date  of  discount  until  the  date  of  maturity. 

267.  When  a  note  or  bond  is  paid  in  part  the  fact  is  acknowledged  by  the 

holder  by  his  writing  the  date  of  payment,  the  sum  paid,  and  his 
signature  on  the  back  of  the  note  or  bond.  This  is  called  an 
indorsement. 

270.  The  small  wheels  under  the  front  of  a  locomotive  engine  are  called 

pilot  wheels,  or  leaders.  The  large  wheels  are  called  drivers. 
The  smaller  wheels  just  behind  the  drivers  are  called  trailers. 

271.  Tractive  force  is  pulling  (or  drawing)  force. 

272.  Replacing  a  letter  (in  an  equation)  by  a  number  is  called  substituting 

the  number  for  the  letter. 
Performing  the  operations  indicated  in  an  equation  and  obtaining 

the  number-value  of  a  letter  is  called  finding  the  value  of  that 

letter. 
278.  The  surfaces  of  the  cube  meet  each  other  in  edges,  thus  forming 

lines. 
The  corners  are  called  vertices.      A  single  corner  is  a  vertex.     (Cf. 

pp.  186,  292.) 
The  edges  meet  each  other  in  corners  of  the  cube,  thus  forming 

points. 

282.  "When  two  lines  meet  making  the  angles  at  their  point  of  meeting 

(intersection)  equal,    the  lines  are  said  to   be  perpendicular  to 
each  other,  and  each  is  called  a  perpendicular  to  the  other. 
The  angles  thus  formed  are  called  right  angles. 

283.  Lines  which  go  through  the  same  point  are  called  concurrent  lines. 
287.  A  rhomboid  is  a  parallelogram  that  is  not  a  rectangle. 

A  trapezoid  is  a  quadrilateral  having  at  least  one  pair  of  opposite 
sides  parallel. 


SYNOPSIS   OF    DEFINITIONS  387 

PAGE 

289.  A  quadrant  of  arc  is  one  quarter  of  a  complete  circle. 

An  angle  is  the  amount  of  turning  of  a  line  about  a  point  as  a 
pivot.  It  may  also  be  regarded  as  the  difference  of  direction  of 
two  lines. 

A  straight  angle  is  the  sum  of  two  right  angles. 

290.  An  angular  degree  is  one  of  the  ninety  equal  parts  of  a  right  angle. 
A  degree  of  arc  is  one  of  the  ninety  equal  parts  of  a  quadrant. 

A  sextant  is  one-sixth  of  a  complete  circle. 
An  octant  is  one-eighth  of  a  complete  circle. 

291.  A  perigon  is  an  angle  equal  to  one  complete  revolution. 

292.  If  two  lines  intersect  each  other,  two  angles  lying  opposite  to  each 

other  are  called  opposite  or  vertical  angles. 

The  lines  which  include  an  angle  are  called  the  sides  of  the  angle. 
The  point  where  the -sides  meet  is  called  the  vertex  of  the  angle. 

(Of.  pp.  186,  278.) 

293.  An  angle  that  is  smaller  than  a  right  angle  is  called  an  acute  angle. 
An  angle  that  is  larger  than  a  right  angle  and  less -than  a  straight 

angle  is  called  an  obtuse  angle. 

295.  Two  angles  whose  sum  equals  a  right  angle,  or  90°,  are  called  comple- 
mental  angles.  Two  angles  whose  sum  equals  two  right  angles,  or 
180°,  are  called  supplemental  angles. 

304.  The  zenith  is  an  imaginary  point  in  the  sky,  directly  overhead. 
HOC.  Longitude  is  the  distance  in  degrees,  minutes,  and  seconds  (of  arc) 
due   eastward  or  westward  from  a  chosen  meridian  called  the 
prime  meridian. 
Local  sun  time  is  obtained  by  setting  timepieces  at  XII  as  the  sun 

crosses  the  meridian. 
311.  The  date  line  is  the  180th  meridian. 

A  square  of  roofing  is  a  ten-foot  square  (100  sq.  ft.). 
Shingles  are  said  to  be  laid  so  many  inches  to  the  weather  when  the 
lower  end  of   each  course  of  shingles  extends  so  many  inches 
below  the  course  next  above  it. 

314.  A  range  is  a  tier  (or  row)  of  townships  running  north  and  south. 

315.  Such  of  the  north  and  the  west  rows  of  half-sections  of  a  township 

as  do  not  have  exactly  320  acres  each  are  called  lots. 

316.  The  volume  of  any  figure  is  the  number  of  cubical  units  within  its 

bounding  surfaces 

321.  A  straight  line  connecting  two  points  of  an  arc  is  called  a  chord  of 

the  arc. 

322.  A  circle  is  said  to  be  circumscribed  around  a  triangle  and  the  triangle 

is  said  to  be  inscribed  in  the  circle  if  all  the  vertices  of  the  tri- 
angle are  on  the  curve. 

325.  The  longest  side  of  a  right  triangle,  that  is,  the  side  opposite  the 
right  angle,  is  called  the  hypothenuse. 


388  RATIONAL    GRAMMAR   SCHOOL   ARITHMETIC 

PAGE 

327.  The   product  obtained   by  using  any  number  twice  as  a  factor  is 

called  the  square  of  that  number. 

328.  The  square  root  of  a  number  is  one  of  its  two  equal  factors. 

332.  The  cube  of  a  number  is  the  product  obtained  by  using  the  number 

three  times  as  a  factor. 

333.  The  cube  root  of  a  number  is  one  of  its  three  equal  factors. 

334.  In  triangles  having  the  same  shape,  angles  lying  opposite  propor 

tional  sides  in  different  triangles  are  called  corresponding  angles. 
In  triangles  having  the  same  shape,  sides  lying  opposite  the  equal 

angles  are  called  corresponding  sides. 
340.  The  amount  paid  for  insurance  is  called  the  premium. 

The  written    agreement   between  an  insurance  company  and  the 

insured  is  called  a  policy. 
The  amount  for  which  the  property  is  insured  is  called  the  face  of 

the  pc'icy. 
342.  A  tax  is  $    sum  of  money  levied  by  the  proper  officers  to  defray  the 

expen:  :s  of  national,  state,  county,  and  city  governments,  and 

for  pu  -lie  schools  and  public  improvements. 
Assessed  1  aluation  means  the  estimated  value  of  the  property  that  is 

assessed. 

345.  A  stock  (called  also  a  stock  certificate)  is  a  written  agreement  made 

by  a  company  to  pay  the  holder  a  certain  part  of  the  earnings  of 

the  company. 
When  a  stock  company  pays  to  the  holders  of  its  stock  #2  on  each 

§100  of  its  capital  stock,  or  2fo  on  its  stock,  the  company  is  said 

to  be  paying  a  $3  dividend,  or  a  2%  dividend. 
At  par  means  at  its  face  value. 
Above  par  means  more  than  its  face  value. 
Below  par  means  less  than  its  face  value. 
A  bond  is  a  written  agreement  made  by  a  national,  state,  county,  or 

city  government,  or  by  a  company,  to  pay  the  holder  interest  at  a 

stated  rate  on  a  stated  sum  of  money,  called  the  face  of  the  bond. 
A  broker  is  a  man  that  makes  a  business  of  buying  and  selling  stocks 

and  bonds  for  other  people. 
A  broker's  charge  for  his  services  is  called  brokerage. 

346.  Preferred  stocks  are  stocks  which  pay  a  fixed  dividend  before  any 

dividends  are  paid  on  common  stocks. 
Common  stocks  pay  dividends  dependent  on  the  net  earnings  of  the 

company  after  expenses  and  dividends  on  preferred  stock  have 

been  paid. 
Government  2s  are  government  bonds  paying  2%  interest. 

347.  Compound  interest  is  interest  on  interest. 

The    compound    amount    is    the    principal    plus     the    compound 
interest. 


SYNOPSIS   OF    DEFINITIONS  389 

349.  Coupon  notes  are  notes,  attached  to  interest-bearing  notes,  for  the 
amount  of  interest  due  at  each  interest-paying  period. 

353.  An  expression  in  which  the  sign  <  or  >  stands  between  two  numbers 

is  called  an  expression  of  inequality. 

354.  The  signs  =,  < ,  and  >  are  called  relation  signs. 
364.  For  static  and  sliding  friction,  see  pp.  364-5. 
367.  For  freezing  and  boilings  points,  see  p.  367. 

General  Definitions 

Number  is  the  result  of  the  measurement  of  quantity. 

Number  may  also  be  defined  as  the  ratio  of  quantities  of  the  same 

kind. 

Quantity  is  limited  magnitude. 
Arithmetic  is  the  science  of  numbers  and  the  art  of  using  them. 


GENERAL   INDEX 

EXPLANATORY.— The  following  index  has  been  prepared  for  daily  use 
by  young  students  of  various  ages  and  the  principal  purpose  underlying 
its  construction  has  been  to  enable  all  students  to  find  readily  and  quickly 
the  thing  desired,  that  it  may  be  in  use  not  occasionally  but  continually. 
To  this  end  the  entries  have  been  arranged  not  merely  for  the  noun  (or  the 
principal  noun)  but  also,  in  many  cases,  for  other  words  of  the  phrase, 
thus  making  the  index  usable  also  by  students  to  whom  grammatical  dis- 
tinctions are  not  entirely  familiar. 

In  reading,  the  phrase  after  the  comma  (where  one  occurs)  is  to  be 
read  first;  thus  "addition,  short  methods  for"  is  read  "short  methods 
for  addition." 

The  references  are  to  pages.  Where  two  page  references  are  sepa- 
rated by  a  dash  the  meaning  is  "and  included  pages;"  thus  27-30  means 
pages  27,  38,  29,  30. 


Above  par,  345 

Account,  forms  of,  9,  28-30.  61,  62.  94-102 

Accounts,  bills  and,  94-102 

Accounts,    farm,    28-30,    61,    62 

Accounts   rendered.  97 

Acquired     territory     of     United     States, 

area  of,  252 
Acre,  227 
Acres  covered  by  one  mile  of  furrow  of 

given  width,  number  of,  171 
Acute  angle  (and  figure),  293 
Addend,   36 

Addition,  checking,  38,  42.   374 
Addition  of  angles,  293-297 
Addition   (of  common  numbers),  35-40 
Addition  of  decimals,  see  also   decimals, 

203,  204 

Addition  of  fractions,  sec  fractions 
Addition    of    fractions    having    common 

fractional   unit.   154 
Addition,    short   methods    for.    374 
Additional   problems  on  town   block   and 

lots,  91,  92 
Admission    of    states,     territories,      etc., 

dates  of,  53 

Air,  pressure  of,  133.  134 
Algebraic  phrases.  185.  186 
Almanac,  250.  251 

Altitude,  see  also  elevation  and  height 
Altitude  of   parallelogram.   116 
Altitude  of  trapezoid.   117 
Altitude  of  triangle,  117 
Amount,  36.  258 
Amount,    compound,   347 
Angles,  289 


Angle   (and  figure),  acute,  293 
Angle   (and  figure),  obtuse,  293 
Angle    (and  figure),   right,   282,   293 
Angle    (and  figure),   straight,   289 
Angle  and  of  arc,  degree  of,  290,  293 
Angle  and  of  arc,  minute  of,  291,  293 
Angle  and  of  arc,  second  of,  291,  293 
Angle.  s'.deg  of,  292 
Angle,  vertex  of,  186.  292 
Angles,    addition    of,    293-297 
Angles  and  arcs,  measuring.  289-293 
Angles     (and    figure),    opposite    or    ver- 
tical, 292 

Angles,   cpmplemental,   295 
Angles,  difference  and  sum  of,  293-297 
Angles,  division  of,  186 
Angles    formed    by    parallel    lines    and .  a 

line  cutting  them,  292 
Angles  of  hexagon,   sum  of.   296 
Angles  of  u-gon,   sum  of.   297 
Angles  of  octagon,  sum  of.  297 
Angles  of  parallelogram,   sum  of.   296 
Angles    of    quadrilateral,    sum    of.    292, 

296 
Angles  of  right   triangle,   sum   of  acute, 

296 

Angles    of    similar    triangle,    correspond- 
ing.  334 

Angles  of  triangle,  sum  of.  292,  295 
Angles,  subtraction  of.  294 
Angles,  sum  and  difference  of.  293-297 
Angles,    supplemental.   295 
Angular  degree.  290.  293 
Angus  cattle,  weights  and  prices  of,  213 
Animals,  habits  of,   15 


390 


GENERAL   INDEX 


391 


Annually,   258 

Apothecaries'  weight,  226 

Apparent   (sun)    noon,   clocktime  of,  304 

Applications  of  percentage,   340-349 

Applications     of     proportion,     practical, 

197-200 
Applications  to  transportation  problems, 

see  also  tractive  force,  270-275 
Arabic  notation,  34 
Arabic  numeral,  34,  35 
Arc,  104,  289 

Arc,  degree  of  angle  and  of,  290,  293 
Arc,  minute  of  angle  and  of,  291,  298 
Arc,  second  of  angle  and  of,  291,  293 
Arch,  figure  of  stone,  140 
Arcs,   measuring  angles  and,  289-293 
Are   (unit  of  land  measure),  238 
Area,    see    also    measuring    surface    and 

mensuration, 

Areas  of  common  forms,   138-140 
Areas  of  countries,  82 
Ascending,   reduction,   230,   232 
Assessed  valuation,  342 
Assessment,   lighting,   92 
Assessment,  paving,  304 
Assessment,  sidewalk,  92 
Assessment,  special  water,  91,  92 
Average,  10,   19,  22,  156 
Avoirdupois  weight,   226 

Bale  (unit  of  paper  measure),  229 

Bale  of  cotton,  weight  of,  171 

Ball,  velocity  of  cannon,  89,  216 

Ball,   velocity  of  rifle,  216 

Bank  discount,  266 

Barometer  (and  figure),  sec  also  weath- 
er, 26-28 

Barometer,  range  of,  28 

Barometer,  reading  of,  26 

Barometer  readings,  table  of,  27 

Barrel,  number  of  gallons  in,  48,  235 

Base    (in  percentage),   245 

Base  line,  314 

Base  of  isosceles  triangle,  108 

Bases  of  trapezoid,  117 

Basins,  areas  of  river,  253 

Battleships,  data  of,  40 

Beans,  capacity  for  absorbing  water, 
205,  206 

Below  par,  345 

Bicycle    (and  figure),   gear  of,   222,   223 

Billion,  32  and  footnote 

Bills  and  accounts,  94-102 

Birds,  speeds  of,  87,  216 

Bisect,   107 

Bisector,   figure   of  perpendicular,   107 

Bisector,  perpendicular,  186 

Block  and  lots  (and  figures),  town,  2-4, 
91,  92 

Blocks,  figure  of  city  streets  and,  116 

Board  foot,   12,  234 

Board  measure,   see  measuring  wood,  12 

Body,  average  surface  of  human,   134 

Body,  daily  requirement  of  water  and  of 
solid  food  for  human,  153 

Body,  normal  temperature  of  a  man's, 
372 

Boiling   point,    367 

Boiling  temperature,   368,  372 

Bond,  345 


Bond,  face  of,  345 

Bond  quotations  and  transactions,  stock 

and,  345 

Bonds,   stocks  and,   345-347 
Book,   figure  of,   194 
Breeze,  kinds  of,  19 
Brick  wall,  figure  of,  313 
Brick  wall,  kinds  of,   4,  footnote, 
Brick  work,  3-6,  92,  93,  313 
Broker.   345 
Brokerage,  345 
Bulk,  see  volume 

Bundle    (unit  of  paper  measure),  229 
Bushel,  228 
Bushel,    number   of   cubic   feet    of   grain 

in,  118,  215 
Bushel     of    various     articles,     table     of 

weights   of,   228 
Bushel,  Winchester,  228 
Butcher's  price  list,  8 


Cables,   data   concerning,   364 
Calculations,  methods  of  shortening  and 

checking,  373-378 
Calendar,    Gregorian,   229 
Calendar,  Julian,  229 
Calendar  month,   229 
Calendar  year,  229 

Cancellation,  84,  88,  89,  168  and  note. 
Cannon  ball,  velocity  of,  89,  216 
Capacity,  69. 

Capacity,  measuring,  118,  119,  228,  238 
Carrier  pigeon,  speed  of,  216 
Cars,      engines      and      tenders,     heights, 
lengths  and  weights  of  railroad,  20, 
76,  135,   136,  271 
Cash    rent,    14 

Casting  out  the   nines,  42,  65,  66,  87 
Cattle  at  Chicago   Stock  Yards,   receipts 

of,  49. 
Cattle,    weights    and    prices    of    Angus, 

213 

Cattle,    weights    and    values   of,    68 
Cavetto  molding,  825 
Cellar  excavation,   5 
Cent,   225 

Center  of  circle,   104 
Centesimi,   226 
Centi-,   237 

Centigrade  thermometer,  367-373 
Centime,  226 
Central  time,  309 
Centre.     See    center 
Century.  229 
Certificate,  stock,  845 
Chain  ;  aZso  square  chain,  227 
Chair,   figure  of,   194 
Change  of  notation,  35 
Checking  calculations,  methods  of  short- 
ening and,  373-378 
Checking  calculations  : 

Addition,   38,   42,   374 

Casting  out  the  nines,  42,  65,  66,  87 

Division,   86,   87,   377 

Multiplication,  65,  66,   375-377 

Square   root,   378 

Subtraction,  375 
Chest  expansion,   112 
Chest  measure,  17 


392 


RATIONAL   GRAMMAR    SCHOOL    ARITHMETIC 


Chicago  ^tock  Yards,   receipts  of  cattle 
Chord  (of  a  circle),  321 


Circle,  area  of,  220,  221 
Circle,   center  of,    104" 
Circles,    concentric,    104 
Circulate,   219 
Circulating  decimal,  219 


.^.  <*M.civni,     jj_o,    ^J.»i,     ^;TO 

'Cl2115erenCe  t0  diameter-  ratio  of,  214, 

Circumferences    of    various    things.     ,SYe 

olso    diameters,    etc      68     114     ifin 

214    215    221  AOW, 

Circumscribed,   322 

Cities,  populations  of,  42    55 

City  streets  and  blocks,  figure  of    116 

(leaning  upon  death  rate,  effect  of  street, 

Coal,  hard  vs.  soft,  8 

of    cubic    feet    of 


of    continents,    lengths    of, 

Coffee  imports  of  United  States,  41,  68, 
*-_LO 


Colors,    rates    of    vibration    for     34 
Commerce,    52-55 
Commission,    254,    255 
Common  denominator,  146   157 
Common  fraction.     See  fraction 
Common  fraction,    reduction   of'  decimal 

Common^fra'cfhm    to    decimal,    reduction 

Common  multiple,  158 

Common  stock,    346 

Common  uses  of  numbers,  133-140 

Common  year,    229 

Comparison  of  prisms,  280,  281 

Comparison  of  sail  areas  of  yachts,  224 

Compasses    (and    figure),    pen-foot    and 

pin-foot   of,   104 
Complemental  angles,  295 
Complex  decimal,  203 
Complex  fractions,  182,  183 
Composite  number,   149 
Compound  amount,  347 
Compound  denominate  numbers,  2°5-24'> 
Compound  interest,    347-349 
Concentric   circles,   104 
Concrete  unit,   225 
Concurrent   lines,   283-285 
Cone,    model    and   development    of    right 

circular,   318 

Cone,  volume  of  right  circular,  318 
Consignment  254 
Constructions.      (See    also     constructive 

geometry   and   models   and   develop- 

ments) : 


Dra  icings  : 

I.  Circle  with  given  radius,  104 

TT'T  LTne  equal  to  given  Hue, KM*] 

Tl""    '    "-    sum   of   two 

105 


V.  Line  equal  to  two,  three  or  four 
times  given   line,   106 

VI  and    I.   Divide    given    line    into 
107,   iff  Pai'tS    (bisect  it}'  1<*° 

VII  Equilateral   triangle  with   each 
side  equal  to  given  line,  108 

viii.     Isosceles  triangle  with   sirip« 
equal  to  two  given  lines    108 
*     *°Hlene  *rlangle  with  sides  equal 
to  three  given   lines,  109 

X.  Three-lobed    figure    in    circle    of 
given   radius,   109,   110 

in  circie  °f 


II.  Divide    given     angle     into     two 
equal  parts  (bisect  it),  186 

I.  Line  parallel  to  given  line,  187 
Y£  lne  Parallel    to  given   line   and 

TT1  through  given  point,  188 

[II.  Line  parallel  to  given  line  and 

Tv^rM?  glven  P°int<  188 
IV.  Divide    given     line     into 
equal  parts   (trisect  it),  188 

3°'an6°  '  and  4° 


,  line 
Through  given  point   on  given 

VTY|ieVrEerpei?dicular  ^   line,   192 
VI 11.  Through    given    point    out    of 
gmm  line,  perpendicular   to   line, 

I.  Perpendicular  to  given  line  from 
given  point  on  it,  276 

II.  Perpendicular     to     given      line 
from   given   point   out  of  it,   276; 

^qu^-e  in  circle  of  given  radius-, 

Greasings    (paper-folding)  : 

I.  Perpendicular      to      given      line 
through  given  point  on  it    282 

II.  Bisect  an  angle    283 

TV'  Shl'el  n°n-Para»el   lines,   283 
ooo  three  angles  of  triangle 

^OO 

V.  Bisect  given  line.  284 

VTT    rft,iare  and  its  diag°nals,  284 
VII.  Three       perpendiculars       from 
of   triangle    to    opposite 


VIII.  Three   perpendicular   bisectors 
of  sides  of  triangle,  285 

.  285 


Find  sum  of  two  given  agles,  294 
294       eren°e  °f  tW°  given 


GENERAL    INDEX 


393 


Find   sum   of  angles  of   scalene   tri- 
angle,  295 
Find  sum  of  angles  of  right  triangle, 

29G 
Find   sum   of  acute  angles   of  right 

triangle,  296 
Find  sum  of  angles  of  quadrilateral, 

296 

Find  sum  of  angles  of  hexagon,  296 
Find  sum  of  angles  of  octagon,  297 
Find  sum  of  angles  of  n-gon,  297 
Find  relations  of  parts  of  rectangle 

with  a  diagonal,  297 
Find  relations  of  parts  of  parallelo- 
gram with  a  diago.nal,  297 
I)r<i ir in<jn  : 

I.  Centre   of  given   arc,   321,   322 

II.  Bisect  given  arc,  322 

III.  Circumscribe   circle   about  equi- 
lateral triangle,   322 

IV.  Trefoil,  322 

V.  Quatrefoil,   322 

VI.  Designs  of  figure  215,  323 

VII.  Sixfoil,  323 

VIII.  Five-point   star   in   circle,    323 

IX.  Regular  pentagon  on  given  line 
as   side,   323 

X.  Cinquefoil,  324 

XI-X1II.   Designs  of  figures    220-223, 
324 

XIV.  Designs    for    moldings    of    fig- 
ures 224,  225,  325 

XV.  Right  triangle,  325 

XVI.  Right    triangle    having    given 
hypothenuse,  325 

XVII.  Right     triangle     with     given 
sides   of   any    right   triangle,    326 
326 

Cuttings  : 

XVIII.  Find   relation   of  squares  of 
sides    of    isosceles    right    triangle, 
326 

XIX.  Find    relation    of    squares    of 
sides  of  any  right  triangle,  326 

Draicin})  : 

Square    roots    of    numbers    and    of 
products,    332 

Constructive  geometry.  Sec  also  con- 
structions, 104,  111,  276-299,  321- 
327 

Copeck,  226 

Cord  (unit  of  fire-icood  measure),  227 

Cord  foot,  227 

Corresponding  angles  of  similar  triangles, 
334 

Corresponding  sides  of  similar  triangles, 
334 

Cost  of  living,  7-10,  99-102 

Cotton,    weight   of  bale,   171 

Counting,    229 

Counting,  measurement  by,  229 

Countries,  areas  and  populations  of,   82 

Couplets,  first  and  second,  196 

Coupon  notes,  349 

Creasing  paper.  See  also  constructions, 
282-285 

Critical  temperatures  of  gases,  372 

Crossing,   figure  of  street,   91 

Crow,  speed  of,  87,  216 

Crown,  225,  226 


Cuba,  area  of,   49 
Cuban  school  data,  40 
Cube,  model    and   development   of    three- 
inch,  278 

Cube,  of  number,  332 
Cube  root  of  number,  333 
Cube  roots,  cubes  and,  332,  333 
Cubes  and  cube  roots,  332,  333 
Cubes  of  units  and  tens,  table  of,  332 
Cubic  feet  of  hay  and  coal  in  ton,  num- 
ber of,  215 

Cubic  foot,   number  of  gallons  in,   271 
Cubic  foot  of  various  things,  weights  of, 

88,   119,   217,   218 

Cubic  foot  of  water,  weight  of,  175,  271 
Cubic  inch  of  steel,   weight  of,  119 
Cubic  inch  of  water,   weight  of,  174 
Curves,  plotted,  23,  24,  27,  126,  128,  272, 

304,  365,  372 

Cylinder,  lateral   surface  of  right   circu- 
lar, 317 
Cylinder,  model      and      development     of 

right  circular,  317 
Cylinder  of  engine,  figure  of,  221 
Cylinder,  volume  of  right  circular,  317 
Cyma  recta  molding,  325 


Dairying.     See  also   milk   and  milkings, 

Data  for  individual  work,  55 

Date  line,  311 

Date  of  maturity,  265 

Dates    of    admission    and    population    of 
states,   territories,   etc.,   53 

Day,  229 

Day,  mean  solar,  124,  229 

Death  rate,  effect  of  street  cleaning  up- 
on, 19 

Death  rate  in  New  York  City,  1886-1896, 
19 

Deci-,  237 

Decimal,  202 

Decimal,  circulating,  219 

Decimal,  complex,  203 

Decimal  fraction.     See  decimal  202 

Decimal,  mixed,  203 

Decimal,  non-terminating.  219 

Decimal  notation,  33,  200,  201 

Decimal  places,  number  of,  208 

Decimal  point,  200 

Decimal  point  in  product,  208,  209 

Decimal,  pure,  203 

Decimal,  reduction    of   common    fraction 
to,  218,  219 

Decimal,  simple,  203 

Decimal  to    common    fraction,    reduction 
of,   202,   203 

Decimals,  200-224 
Notation,  200.  201 
Numeration,  201,  202 
Reduction  of  decimal  to  common  frac- 
tion. 202.  203 
Addition,  203,  204 
Subtraction,  205,  206 
Multiplication    (pointing  off  the  prod- 
uct),  208,   209 
Division,  210-213  : 

Of  decimal  by  integer,  210-212 
By  decimal,  212,  213 


394 


RATIONAL    GRAMMAR   SCHOOL    ARITHMETIC 


Reduction  of  common  fraction  to  deci- 
mal, 218,  219 

Decimals,  addition  of.  203,  204 
Decimals,    division    of.       See     decimals, 

202 

Decimals,  notation  of,  200,  201 
Decimals,  numeration  of,  201,  202 
Decimals,  principles    of.     Sec    principles, 

etc. 

Decimals,   subtraction  of,  205,  206 
Degree,  angular,  290,  293 
Degree  of  angle,  290.  293 
Degree  of  arc,  290,  293 
Degree  of    longitude    at    equator,    length 

of,  174 
Deka-,  237 

Denominate  number,  225 
Denominate  numbers,  compound,  225-242 
Denominator,  145 
Denominator,  common,  146.  157 
Denominator,  least  common,  157 
Depth,  greatest   ocean,  81 
Depths  of  lakes,  254 
Descending,  reduction,  230,  232 
Description  of  land.  314-316 
Development,  115,  279 
Developments.     For  list  see  models,  etc. 
Diagonal,   139,   296 
Diagonal   divides   figure,   how,   189,    296, 

297 

Diameter,  104,   113 
Diameter,  ratio  of  circumference  to,  214, 

215 

Diameters  of  earth,  216,  319,  321 
Diameters  of  heavenly  bodies,  321 
Diameters  of  various  things.  See  alto 

circumferences,   etc.,    114,   214,   215, 

221-223 
Difference,  47 

Difference  of  angles,  sum  and,  293-297 
Differences    of    lines,    products    of    sums 

and,   297-299 
Digit  or  figure,  31 
Digit,  name   value  of,  31 
Digit,  place  value  of,  31 
Dime,  225 

Dimensions  of  trapezoid,  118 
Discount,   255,   266 
Discount,  bank,  266 
Discount,  trade,  255,  256,  343,  344 
Discount,  true,  266 
Discounting  notes,  266,  267 
Displacement  of  vessel,  40 
Distance,  measuring,  112,   114,  226,  227 
Distances  from  Chicago  to  various  cities, 

59,   67,    134,    216,   362 
Distances  from    Washington    to    various 

cities,  66 
Distribution    of    population    of     United 

States,  1900,  43-45 
Distribution    of    sun's    light    and    heat, 

299-305 

Dividend,  74,  345 

Divisibility    of    numbers.     See    also    fac- 
tors,    prime     and     composite,     and 

greatest  common  divisor,   85,   86 
Division  and     multiplication     compared, 

73-75 
Division  and  subtraction    compared,    72, 

73 


Division   by   multiples   of   10',    82-84 

Division,   checking,  86,  87,  377 

Division,  greatest  common  divisor  by 
successive,  151 

Division,  long,  77,  78 

Division   (of  common  numbers),  72-93 

Division  of  decimals.     See  decimals 

Division  of  fractions.     See  fractions 

Division  of  lines  and  angles,  186 

Division,  short,  75,  76 

Division,  short  methods  for,  82-86,  201, 
377 

Divisor,  74 

Divisor,  greatest  common.  Sec  also  great- 
est common  divisor,  147 

Dollar,  111,  225 

Door,  figure  of,  193 

Double  eagle,  225 

Dozen,  229 

Dram,  226 

Drivers  of  locomotive,  270 

Dry  gallon,  liquid  and,  228 

Dry  measure,  228 

Duck,  speed  of  wild,  216 


Eagle  :  half-eagle  ;  quarter-eagle  ;  double- 
eagle,  225 
Earth,    area    of :    land,    area    of ;    water, 

area   of,   252 

Earth,  diameters  of,  216,  319,  321 
Earth,  mean  diameter  of.  319.  321 
Earth,  mean  radius  of,  33,  320 
Earth,  velocity  of,  25,  89 
Eastern  time,  309 
Echinus,  or  ovolo,  molding.  825 
Electricity,  velocity  of,  216 
Elevation.     See  also  altitude  and  height 
Elevations  and  heights  of  mountains,  51, 

81,  253 

Elevations  of  lakes,  49,  51,  254 
Elevations  of  Weather  Bureau   stations, 

249 
Engines    and    tenders,    heights,    lengths 

and  weights  of  railroad  cars,  20,  70, 

135,  136.  271 

English  notation,  32  and  note 
Equal    parts,    fraction   as    ratio    and   as, 

143-146 

Equation,  first  and  second  sides  of,  103 
Equation,   principles  for  using.     »S'< c  <ilmt 

principles,  etc.,   103.   353-356,  370 
Equation,   uses  of,  352,  364-373 
Equilateral    triangle    (and    figure),    108, 

109 
Equivalent     readings    of    thermometers, 

371-373 
Equivalents,    metric    and   United    States, 

237-239 

Equivalents  of  longitude  and  time,  306 
Even  number,  149 
Expansion,  chest,  112 
Expense  account,  family,  99-102 
Expenses  of  living,  8,  10,  99-102 
Exponent,   34,   149 
Export  trade,  1891,  1901,  United  States, 

54 

Express  trains,  speed- of,  88.  89 
Expression  of  inequality,  353 
Extremes,  182,  196,  197 


GENERAL 


395 


Face,  265 

Face  of  bond,  345 

Face  of   policy,   340 

Face  value,  265 

Factor,  63,  148 

Factor,  prime,  149 

Factors,     greatest    common     divisor     by 

prime.    150-151 

Factors,  multiplication  by,  63,  64 
F'actors,  order  of,  170 
Factors,  prime  and  composite.     Sec  also 

divisibility  of  numbers,  148,  149 
Factory    life    upon     growth,     effect    of, 

207 

Fahrenheit,  thermometer,  122,  367-373 
Falcon,  speed  of,  216 
Family   expense   account,   99-102 
Farm  accounts,  28-30,  61,  62 
Farm,  fencing,  11-13,  139 
Farm,   figure  of,   11 
Farthing,   225 

Fence  wire,  cost  and  weight  of,  11,  12 
Fencing  farm,  11-13,  139 
Fields,  areas  of,  13,  14 
Figure  or  digit,  31 
Filler,  225 

Finding  the  value  of  (phrase),  -272 
Fineness  of  United   States  gold  and  sil- 

ver coins,   233 

Fire-wood  by  cord,  measuring,  227 
Flat   prism,    model   and  development   of, 

280 

Floor  plans  of  house,  figure  of,  7 
Folding    paper.     See   also    constructions, 


Food,  prices  of,  8 

Foot,  board,  12,  234 

Foot,  cord,   227 

Foot,  number  of  gallons  in  one  cubic,  271 

Foot  ;   also   square   and  cubic   foot,    226, 

227 
Foot    of    compasses,    pen-foot    and    pin- 

foot,    also    square    and    cubic    foot, 

104,  226,  227 
Force.     See    also    friction    and    tractive 

force 

Forces,  joint  effects  of,  183-185 
Forms,   areas  of   irregular   but    common, 

138,  298,  299 
Forms,  figures  of  irregular  but  common, 

138,   288,  298,   299 
Forms  of  account,  9,  28-30,  61,  62,  94- 

102 

Foundations  of  house  (and  figure),  5,  6 
Fraction    as    ratio    and    as    equal    parts, 

143-146 

Fraction,  common.     See  fraction 
Fraction,  decimal.     See  decimal 
Fraction,   improper,   161 
Fraction,  lowest  terms  of,  147,  151 
Fraction,  proper,  161 
Fraction,   reduction   of  decimal   to    com- 

mon, 202,  203 
Fraction,  terms  of,  145 
Fraction   to    decimal    reduction   of   com- 

mon, 218,  219 
Fractional    numbers,    multiplication    by, 

66 
Fractional  unit,  113,  144 


Fractions,  141-200  : 

Introduction     (ratio     and    proportion, 
141-146 

Ratio  (measure),  141-142 
Proportion,  142,  143 
Fractions    as    ratios    and    as    equal 
parts,    143-146 
Fraction    (simple),    146-200 

Reduction  to  higher,  lower  and  low- 
est  terms,    146-148 
Factors,   prime  and  composite,   148, 

149 
Greatest   common   divisor   by   prime 

factors,  150,,  151 

Greatest  common  divisor  by  succes- 
sive division,   151 
Greatest    common    divisor    of    two 

given  lines,  152 

Addition   of    fractions   having    com- 
mon fractional  unit,  154 
Addition  of  fractions  easily  reduced 
to     common     fractional     unit,     154, 
155 

Multiples  and  least  common  multi- 
ple, 158-161 

Reduction  of  whole  and  mixed  num- 
bers to  fractions,  and  inversely, 
161-163 

Addition,  163-165 
Subtraction.  165-167 
Multiplication,  167-175 

Of  fraction  by  whole  number,  167, 

168 

Of  mixed  number  by  whole  num- 
ber, 169 

Of  whole  number  by  fraction,  170 
Of  fraction  by  fraction,  172,  173 
Of  mixed  number  by  mixed  num- 
ber, 173-175 
Division,  175-183 

Of  fraction  by  whole  number,  175- 

177 

Of  mixed  number  by  whole  num- 
ber,   177,    178 
Of  any  number  by   fraction,   178- 

182 

Fractions    (complex),   182-183 
Fractions,    addition    of.     Sec    also    frac- 
tions,  163-165 

Fractions,   complex,   182,   183 
Fractions  easily  reduced  to  common  frac- 
tional unit,  154,  155 
Fractions  having  common  fractional  unit, 

154 

Fractions,  introduction  to,  141-146 
Fractions,  principles  of.     See  principles, 

etc. 

Fractions,  reduction  of.  See  also  deci- 
mals and  fractions,  146-148,  151, 
154,  155,  161-163,  202,  203,  218- 
219 

Fractions,  subtraction  of.     See  fractions 
Fractions    to   higher,    lower    and    lowest 

terms,  reduction  of,  146-148 
Fractions  with   common   fractional  unit, 

addition  and  subtraction  of,  154 
Franc,  225,  226 
Freezing  point,   367 
Freezing  temperatures,  368 
Freight  and  passenger  trains,  134-136 


396 


RATIONAL    GRAMMAR    SCHOOL   ARITHMETIC 


French  notation,  32  and  note 
Friction   (force  of),  364 
Friction,  law  of  sliding,  367 
Friction,  sliding  and  static,  364-367 
Fulcrum,  89 

Furnishings,   house  and,   4-7,  92,  93 
Furrow  of  given  width,  number  of  acres 
covered  by  one  mile  of,  171 

Gain  and  loss,  246-248 

Gale,  19 

Gallon,  dry  gallon  and  liquid,  228 

Gallon  of  milk,  number  of  pounds  in 
one,  214 

Gallons  in  barrel,  number  of,  48,  235 

Gallons  in  one  cubic  foot,  number  of 
(liquid),  271 

Gases,  critical  temperatures  of,   372 

Gate,  figure  of,  194 

Gear  of  bicycle  (and  figure),  222,  223 

Geographic  mile  or  knot,  234 

Geography,  51,  52,  80-82,  251-254 

Geometry,  constructive.  See  also  con- 
structions, 104-111,  276-299,  321- 
327 

German  notation,  32  and  note 

Gill   (unit),  228 

Globe,  figure  of,  310 

Gold  and  silver  coins,  fineness  of  United 
States,  233 

Gold,  value  of  one  ounce  of,  67,  121 

Gold,  value  of  one  pound  of,  171 

Government  2s,   etc.,    346 

Graduation  of  thermometers,  367-370 

Grain  (unit  of  weight),  226 

Grain  rent,  14 

Grain,  238,  239 

Gravity,  specific,  217,  218 

Great  gross,  229 

Great  Pyramid,  dimensions  of,  320 

Greatest  common  divisor,  147 

Greatest  common  divisor  by  prime  fac- 
tors, 150,  151 

Greatest  common  divisor  by  successive 
division,  151 

Greatest  common  divisor  of  two  lines 
(geometrically),  152 

Gregorian   calendar,   229 

Grocer's  price  list,  8 

Grocery  clerk,  problems  of,  98 

Gross,   229 

Growth,  effect  of  factory  life  upon,  207 

Growth  of  boys  and  girls  in  height  and 
in  weight,  18,  127,  128,  206,  207 

Growth  of  trees,  155-157 

Growth  of  trees,  lateral,  156,  157 

Growth  of  trees,  terminal,  156.  157 

Growth  of  United  States,  territorial,  252 

Guinea,   225 

Habits  of  animals,   15 

Hawk,  speed  of,  87,  216 

Hay  and  of  coal  in  ton,  number  of  cubic 

feet  of,  215 
Heat,  distribution  of  sun's  light  and.  See 

also  weather.  299-305 
Height.     See  also  altitude  and  elevation. 
Height  and  weight,  growth  of  boys  and 

.     girls  in,   18,  127,   128,  206,  207 
Height  to  lung  capacity,  relation  of,   17 


Heights  of  boys  and  girls  working  and 
not  working  in  factories,  207 

Heights  of  mountains,  elevations  and, 
51,  81,  253 

Heights  of  persons,  18,  127,  128,  206 
207,  216 

Hekto,  237 

Helios,  300. 

Heller,  226 

Hexagon  (and  figure),  regular,  110 

Hexagon,  sum  of  angles  of,  296 

High  land,  ratio  of  low  land  to,  252 

Horse-power,   216 

Horse,  speed  of,  216 

Hour,    229 

Hour  circle  of  sun,  310 

House  and  furnishings,  4-7,  92,  93 

House  and  roof,  figure  of,  71,  313 

House,  figure  of  end  of,   193 

House  plans,  5-7,   92,   93 

House,  walls  of,  4-6 

Hundred-weight,  226 

Hundredth,   201 

Hundreths,  measuring  by.  See  also  per- 
centage and  ratio,  129-131 

Hurricane,  19 

Hypothenuse,   325 

Immigration    into    United    States,    1900, 

1901,  41,  76,  77 
Imports    of    coffee   ,into    United    States, 

41,  68,  213 
Imports  of  molasses  into  United   States, 

Imports  of  tea  into  United  States,  41, 
68,  213 

Improper  fraction,  161 

Inch  precipitation,  69,  203 

Inch  ;  also  square  and  cubic  inch,  226, 
227 

Independence  Hall,  Philadelphia,  figure 
of,  278 

Index  notation.  34,  149 

Indorsement,  267 

Industrial  products,  1897,  1902,  values 
of,  54 

Inequality,  expression  of,  353 

Inscribed  square,  192 

Insurance,  340,  341 

Insurance  policy,  340 

Integer.  161 

Interest,  131,  132,  258-264 

Interest,  compound,  347-349 

Interest,  percentage  and,  129-131,  242- 
269 

Interest,  simple,  131,  132 

Interest  table,  263 

Intersection,  105 

Introduction,  general.   1-30 

Introduction  to  fractions,  141-146 

Invert,  180 

Irregular  forms  by  weight  and  propor- 
tion, areas  and  volumes  of,  139,  140 

Isosceles  triangle   (and  figure).  108,  109 

Isosceles  triangle,  base  of,  108 

Jib,  2E4 

Joint  effect  of  forces,  183-185 
Julian  calendar.  229 
Jupiter,  diameter  of,  323 


GENERAL    INDEX 


397 


Kilo-,  237 
Kite,  figure  of,  194 

Knot   or  geographic   mile    (nautical  unit 
of  distance),  234 


Lakes,  areas  of,  254 

Lakes,  depths  of,  254 

Lakes,  elevations  of,  49,  51,  254 

Land    (and  figure),  section  of,   125,   227, 

315 

Land  area  of  the  earth,  252 
Land  areas  of  divisions  of  United  States, 

211 

Land,  description  of,  314-316 
Land,   figure   of  divided   section   of,   125, 

315 
Land,  figure  of  tract'  of,  11,  38,  91,  116, 

125.    139,    195,    196,    199,    314,    315, 

336-339 
Land,  measuring,  125,  126,  227,  238,  314- 

316 

Land  surveying,  314-316,  335-340 
Lateral  growth  of  trees,  156,  157 
Law  of  sliding  friction,  367 
Leaders  of  locomotive,  270 
Leap  year,  229 

Least  common  denominator,  157 
Least  common  multiple,  159-161 
Leaves  of  trees,  205,  249 
Length,    measuring,    112-114,    226,    227, 

237 
Letters    to    represent    numbers,    use    of, 

349-364 

Level   (and  figure),  A,  300 
Lever  (and  figure),  89-91 
Light    and    heat,    distribution    of    sun's. 

Sec  also  weather,  299-305 
Light    to    travel    from    moon    to    earth, 

time  for,  67 
Light  to  travel  from  sun   to  earth,   time 

for,  67 

Light,  velocity  of,  33,  63,  67,  216 
Lighted  (room),  well,  7,  114 
Lighting  assessment,  92 
Line,  278 
Line,  base,  314 
Line,  date,  311 
Line,  standard  base,  314 
Linear  measure,  226,  227,  237 
Lines,  concurrent.  283-285 
Lines,  division  of,  186 
Lines,  figure  of  parallel,   105,   106,   187, 

292 

Lines,  parallel,  190 
Lines,  products  of  sums  and  differences 

of,  297-299 

Link   (unit)  :  also  square  link,  227 
Liquid  and  dry  gallon,  228 
Liquid  measure,  228 
Lira,  225,  226 
Liter,  238 

Literal  numbers,  57 
Literal   numbers,   subtraction  of,   56 
Live  stock,   a   week's   receipts   and   ship- 
ments of.     See  also  cattle,  50 
Living,  cost  of,  7-10,  99-102 
Living,  expenses  of,  8,  10,  99-102 
Load.     See  tractive  force 
Local   (sun)  time,  306 


Locomotive  (and  figure).  See  also  en- 
gines and  tenders,  270-275 

Locomotive,  drivers  of,  270 

Locomotive,  leaders  of,  270 

Locomotive,  trailers  of,  270 

Locomotives,  lengths  and  weights  of, 
135,  136,  271 

Long   division,    77,   78 

Long  ton,  226 

Longitude  and  time,  306-311 

Longitude  and  time,  equivalents  of,  306 

Longitude  at  the  equator,  length  of  one 
degree  of,  174 

Longitudes  of  observatories,   307,   308 

Longitudes,  table  of,   308 

Loss,  gain  and,  246-248 

Lot  (of  land)   (and  figure),  315 

Lots  (and  figure),  town  block  and,  2-4, 
91,  92 

Low  land  to  high  land,  ratio  of,  252 

Lowest  terms  of  fraction,  147,  151 

Lumber  measuring,  12 

Lumber  sold  by  board  feet,  12 

Lung  capacities,  heights  and  weights  of 
men,  216 

Lung  capacity,  16,  17 

Lung  capacity  to  height,   relation  of,  17 

Maker  (of  a  note),  265 

Manufactured      products,      1897,      1902, 

values  of,  54 

Manufactures,  Chicago,  82 
Manufactures,  New  York  City,   1890,   51 
Map  of  time  belts  of  United  States,  309 
Maps  of  United  States,  43,  309 
Mark,  225,  226 
Marking  goods,  257 
Mars,  diameter  of,  321 
Maturity,  date  of,  265 
Mean,  17,  foot-note 
Mean  solar  day,  124.  229 
Mean  solar  year,  229 
Mean  temperatures,  368 
Means,  182,  196,  197 
Measure,  142 
Measure,  chest,  17 
Measure,  dry,  228 
Measure,  linear,  226,  227.  237 
Measure,   liquid,  228 
Measure,  numerical,  142 
Measure,  surveyors',  227 
Measure,  to,  111 

Measurement,  73,  111-132,  142,  144 
Measurements,  45.  46 
Measurements,  physical,  16-19,   216,  217 
Measures,  metric,  237-239 
Measuring.     See  also  mensuration. 
Measuring  angles,  289-293 
Measuring  arcs,  289-293 
Measuring  brick  work,  3-6,  92.  93,  313 
Measuring  bulk.     See  measuring  volume 
Measuring  by  counting,  229 
Measuring    by    hundredths.         See    also 

percentage  and  ratio,  129-131 
Measuring     by     money.     See     measuring 

value 

Measuring  capacity,  118,  119,  228,  238 
Measuring  distance.  112-114.  226.  227 
Measuring  fire-wood,  125,  126,  227,  238, 

239 


398 


RATIONAL    GRAMMAR    SCHOOL   ARITHMETIC 


Measuring  land,  314-310,  112-114,  220 

Measuring  length,  227,  2K7 

Measuring  lumber,  12 

Measuring  paper,  229 

Measuring  roofing,   311-313 

Measuring  stone,  215,  227 

Measuring  surface,    114-118,   227,   238 

Measuring  temperature.   122,  123 

Measuring  time,  124,  125,  229 

Measuring  value,  225,   226 

Measuring  volume,  118,  119,  227,  238 

Measuring  weight,  120-122,  220,  238,  ii:J9 

Measuring  wood  (not  flre-icood) ,  12 

Median,  186 

Melting  temperatures,  368.  371 

Members  of  equation,  first  and  second, 
103 

Mensuration.  See  also  measuring,  etc., 
311-321 

Mensuration  of  circle,  surface,  220,  221 

Mensuration  of  circumference,  214,  215 

Mensuration  of  cone  (right  circular), 
volume,  318 

Mensuration  of  cylinder  (right  circular), 
lateral  surface,  317 

Mensuration  of  cylinder  (right  circular), 
volume,  317 

Mensuration  of  irregular  form  by  weight 
and  proportion,  139,  140 

Mensuration  of  oblique  parallelepiped, 
volume,  281,  282,  316 

Mensuration  of  oblique  prism,  volume, 
281,  282 

Mensuration  of  pail,  capacity,  16 

Mensuration  of  parallelepiped  (and  fig- 
ure) (rectangular),  volume,  70,  118, 
119,  316 

Mensuration  of  parallelepiped  (oblique), 
volume,  281,  282,  316 

Mensuration  of  parallelogram,  surface, 
116,  117,  189 

Mensuration  of  prism  (and  figure)  (tri- 
angular), volume,  320 

Mensuration  of  prism  (oblique),  volume, 
281,  282 

Mensuration  of  prism  (right),  volume, 
281,  282 

Mensuration  of  prism  (square),  volume, 
118,  119,  281,  282,  316 

Mensuration  of  pyramid  (and  figure) 
(triangular),  volume.  320 

Mensuration  of  ratio  of  circumference  to 
diameter.  214,  215 

Mensuration  of  rectangle,  surface,  45,  46. 
70,  114,  116,  189 

Mensuration  of  rectangular  parallele- 
piped, volume,  70,  118.  119,  316 

l.Iensuration  of  right  circular  cone,  vol- 
ume. 318 

Mensuration  of  right  circular  cylinder, 
lateral  surface  317 

Mensuration  of  right  circular  cylinder, 
volume,  317 

Mensuration  of  right  prism,  volume, 
281,  282 

Mensuration  of  sphere,  surface,  319 

Mensuration  of  sphere,  volume,  321 

Mensuration  of  square-cornered  box,  vol- 
ume, 69,  70,  118 


Mensuration  of  square  prism,  volume, 
118,  119,  281,  282,  816 

Mensuration  of  square,  surface,  114,  189 

Mensuration  of  trapezoid,  surface,  117, 
118 

Mensuration  of  triangle,  surface,  117, 
189 

Mensuration  of  triangular  prism  (and 
figure),  volume,  320 

Mensuration  of  triangular  pyramid  (and 
figure),  volume,  320 

Mercury,  diameter  of,  321 

Meridian,  306,  310,  314 

Meridian,  prime,  306,  310 

Meridian,  principal,  314 

Meteorology.     Sec  also  weather,  248-250 

Meteors,   November,   2*5 

Meter ;  also  square  and  cubic  meter,  236- 
238 

Methods  of  shortening  and  checking  cal- 
culations, 373-378 

Metre.     See  Meter 

Metric  measures,   237-239 

Metric  system  and  equivalents  in  United 
States  system,  236-239 

Metric  system,  history  of,  236.  237 

Metric  ton,  239 

Mile  or  knot,  geographic.  234 

Mile  ;  also  square  mile,  226,  227 

Mile,   statute,  234 

Mileage  and  numbers  of  employees  of 
street  railroads,  77 

Mileage  of  United  States ;  1900,  rail- 
road, 320 

Milk,  number  of  pounds  in  one  gallon 
of,  214 

Milk,  weight  of  one  quart  of,  22 

Milkings,  weights  of.  See  also  dairying, 
21,  214 

Mill,   225 

Mill!-,   237 

Million,  32  and  foot-note 

Minnend,  47 

Minute  of  angle,  291.   293 

Minute  of  arc,  291,  293 

Minute  of  time,  229 

Mixed  decimal,  203 

Mixed  number,  161 

Model  and  development  of  cone  (rigKl 
circular).  318 

Model  and  development  of  cube  (three- 
inch),  278 

Model  and  development  of  cylinder 
(right  circular),  317 

Model  and  development  of  flat  prism, 
280 

Model  and  development  of  oblique  paral- 
lelogram prism,  281 

Model  and  development  of  prism  (flat). 
280 

Model  and  development  of  prism  (ob- 
lique parallelogram),  281 

Model  and  development  of  prism  (right), 
280 

Model  and  development  of  prism 
(square),  279 

Model  and  development  of  prism  (tri- 
angular), 281 

Model  and  development  of  pyramid  (tri- 
argular),  320 


GENERAL   INDEX 


399 


Model  and  development  of  right  circular 

cone,  318 
Model  and  development  of  right  circular 

cylinder.   317 
Model   and  development   of  right   prism, 

280 

Model    and    development   of   roof,   311 
Model  and  development  of  room,  115 
Model  and  development  of  square  prism, 

279 
Model    and    development    of    three-inch 

cube,  278 

Model  and  development  of  tower,  313 
Model    and    development    of    triangular 

prism.  281 
Model    and    development    of    triangular 

pyramid,   320 
Molasses  into  United  States,  imports  of. 

41 

Moldings,  figures  of,  325 
Money,  111 

Money,  kinds  of,  225,  226 
Month,   calendar,   229 
Moon,  diameter  of.  321 
Moon,  mass  of,  34 
Moon,    mean    distance    from    earth,    33, 

199,  216 

Moon,  velocity  of,  89 
Mountain  time,  309 
Mountains,  elevations  and  heights  of,  51, 

81.  253 

Movement  of  wind,  248,  249 
Multiple,  158,  159 
Multiple,  common,  158 
Multiple,   least  common,   159-161 
Multiplicand.  58 

Multiplication  by   factors.   63,   64 
Multiplication  by  fractional  numbers.  66 
Multiplication   by   number  near  10,   100, 

1000,  etc.,  64 

Multiplication  by  10.  100,  1000,  etc.,  64 
Multiplication  br   25,   50,   12  Vi,  75,  500. 

250,   pp.   64,   65 

Multiplication,  checking.  65,  66,  375,  377 
Multiplication    compared,    division    and. 

73-75 
Multiplication  (of  common  numbers),  57- 

72 

Multiplication  of  decimals.  See  decimals 
Multiplication    of    fractions.      See    frac- 
tions 
Multiplication,  short  methods  for,  63-65, 

202,    375-377 

Multiplication  table,  59,  60 
Multiplication  when  some  digits  of  multi- 
plier are  factors  of  others,  65 
Multiplier.   58 
Myria-,  287 

Name  value  of  digit,  31 

Names  of  winds,  19 

National  League,  one  season's  record  of, 

244 

Nature  study,  205,  206 
Negative,  184 
Neptune,  diameter  of,  321 
Net   proceeds,   255 
N-gon,  sum  of  angles  of,  297 
Nines,  casting  out  the.   42.  65,  66,  87 
Non-terminating  decimal,  219 


Noon,  229 

Noon,  clock  time  of  apparent  (sun),  304 

Normal,  17 

North  America,  area  of,  253 

Notation  and  numeration,  31-35 

Notation,  Arabic,  34 

Notation,  change  of.  35 

Notation,  decimal,  32,  200,  201 

Notation,  English,  30  and  note 

Notation,  French,  32  and  note 

Notation,  German,  32  and  note 

Notation,  index,  34,  149 

Notation  of  decimals.  200,  201 

Notation,  Roman,  34,  35 

Notation,  United  States,  32  and  note 

Note,  promissory,  265,  266 

Notes,  coupon,  349 

Notes,  discounting,  266,  267 

Number,  composite,  64.  149 

Number,  denominate,  225 

Number,  even,  149 

Number,  mixed,  161 

Number,  odd.  149 

Number  of  decimal  places,  208 

Number,  prime,  64,  149 

Number,  square  of,    327 

Numbers,  common  uses  of,  133-140 

Numbers,  compound  denominate,  225-242 

Numbers,  divisibility  of,  85,  86 

Numbers,  literal,  57 

Numbers,  periods  of,  32 

Numbers,  reading,  33,  34 

Numbers,  subtraction  of  literal,  56 

Numbers,    use    of    letters    to    represent, 

349-364 

Numbers,  writing,  34 
Numeral,  Arabic,  34,  35 
Numeral,  Roman,  34,  35 
Numeration,   notation  and.  31-35 
Numeration  of  decimals,  201,  202 
Numerator,  145 
Numerical  measure,  142 

Oblique   parallelepiped,  'volume   of,    281, 

282,  316 
Oblique  parallelogram  prism,  model   and 

development  of,  281 
Oblique   prism    (and   figure),   volume   of, 

281.  282 

Observatories,   longitudes  of.  307,   308 
Obtuse  angle  (and  figure),  293 
Ocean  depth,  greatest,  81 
Octagon,  sum  of  angles  of,  297 
Octant.  290 
Odd  number.   149 
One-brick  wall.  4  foot-note 
Operations,  order  of,  169 
Opposite  or  vertical  angles,  292 
Order  of  factors.  170 
Order  of  operations,  169 
Ounce,  226 

Ounce  of  gold,  value  of,  121 
Ovolo,  or  echinus,  molding,  325 
Ovolo,  or  quarter-round,  molding,  325 

Pacific  time,  309 

Pail,  capacity  of,  16 

Paper-folding   (creasing),  282,  285.     See 

also  Constructions 
Paper,  measuring,  229 


400 


RATIONAL    GRAMMAR    SCHOOL    ARITHMETIC 


Par.  above.  345 

Par,  at,  345 

Par,  below,   345 

Parallel  lines,   190 

Parallel    lines    and    line    cutting    them, 

angles   formed   by,   liOU 
Parallel    lines,   figure  of,    105,    106,   187, 

292 

Parallel  ruler  (and  figure),  187.  188 
Parallelepiped,    volume    of    oblique,    281, 

282,   316 
Parallelepiped,    volume    of    rectangular, 

70,  118,  119,  316 
Parallelogram,   287 

Altitude  of,   116 

Area   of,   116,    117,    189 

Compared    (and  figures),  circle  and, 
220 

Figure  of,  115-117,  189,  296,  297 

Properties   of.   297 

Sum  of  angles  of,  296 
Partial  payments.  267-269 
Partial  Payments,  United  States  Rule 

for,  268 

Paving,   3,   4,  91 
Paving  assessment,  3,   4 
Payee.  265 

Payments,  partial,  267-269 
Peck,  228 

Pen-foot  of  compasses,  104 
Penny,  225 
Pennyweight,  226 
Pentagon   (and  figure),   323 
Per  annum.  258 
Per  cent,  15,  130,  242 
Percentage,  129-131,  242-246 
Percentage    and    interest,    129-131,    242- 

269 

Percentage,  applications  of,  340-349 
Perch   of  stone,  215,   227 
Perigon,  291,  293 
Perimeter,  111,  286-288 
Periods  of  numbers,    32 
Perpendicular,  282 
Perpendicular   bisector,   186 
Perpendicular  bisector,   figure   of,    107 
Personal  property,  342 
Persons,  heights  and  weights  of,  18,  127. 

206,  207,  216 
Pfennig,  226 

Phrases,  algebraic,  185,   186 
Physical  measurements,  16-19,  216,  217 
Pi,  value  of.  215 
Pigeon,  speed  of  carrier,  216 
Pilot  wheels  of  locomotive,   270 
Pin-foot  of  compasses,  104 
Pint,   228 

Piston,  figure  of,  221 
Place  value  of  digit.  31 
Places,  number  of  decimal.  308 
Planets,   diameters  of,  321 
Play-house,   figure  of,   193 
Plotted    curves,    23,     24,     27,    126,     128, 

272,   304,   365,   372 
P'otting  observations  and  measurements, 

126-129 

Point   (decimal).  200 
Point   (geometric),  278 
Pointing   off    product    of    decimals,    208, 

209 


Policy,  face  of,  340 

Policy,  insurance,  340 

Population    of     Switzerland,     area    and. 
81 

Population  of  United  States,   1790-1900, 
129 

Population   of  United   States,   1900,  dis- 
tribution of.   43-45 

Populations  of  cities,  42,  55 

Populations   of  countries,   areas  and,   82 

Populations    of    states,    territories,    etc., 
1890,  p.  53 

Populations    of    states,    territories,    etc., 
1900,  p.  44 

Positive,  184 

Positive  and   negative  readings  of   ther- 
mometer, 369,  371 

Pound,  226 

Pound,    comparison    of    avoirdupois    and 
troy,  122 

Pound  sterling,  225 

Pounds  in  one  gallon  of  milk,  number  of 
214 

Power  (of  a  number),  202 

Practical  applications  of  proportion,  197- 
200 

Precipitation  in  various  places,  39.  68-71, 
203,  204,  250.      Sec  also  weather 

Precipitation,  inch,    69,  203 

Preferred  stock,  346 

Premium,  340 

Present  worth.  266 

Pressure  of  air,  133,   134 

Pressure  of  wind,  19,  20 

Price.     Sec  marking  goods 

Prices  of  provisions,  8 

Prime  factor,  149 

Prime  meridian.  306.  310 

Prime  number,  64,  149 

Prime  to  each  other   (phrase),  148 

Principal,   254.   258 

Principal  meridian,  314 

Principles   for   using    the   equation,    354- 

356,  370 

I  354.  II  354,  III  355,  IV  356,  V  356, 
VI  370 

Principles  of  decimals,  203,  209,  213  : 
I  203,    II   209,   III  213 

Principles    of    fractions,    147,    150,    154, 

160,   162,  163.  168,  176,  180: 
I  147,  II  150,  III  154.   IV  160,   V  162, 
VI  162,  VII  163,  VIII  168,   IX  176, 
X  180,  XI  180 

Prism    (and  figure),    volume  of   triangu- 
lar, 320 

Prism,    figure   of    oblique    (oblique   pile), 
282 

Prism,  figure  of  right   (square  pile),  281 

Prism,    model    and    development    of   flat, 
280 

Prism,   and  development  of  oblique  par- 
allelogram, 281 

Prism,  model  and  development  of  right, 
280 

Prism,  model  and  development  of  square, 
279 

Prism,   model  and  development  of  trian- 
gular, 281 

Prism,  volume  of  oblique,  281,  282 


GENERAL    INDEX 


401 


Prism,  volume  of  right,  281.  282 

Prism,  volume  of  square,  118,  119,  281 
282,  316 

Prisms,  comparison  of,  280 

Problem,  statement  of,  58 

Problems  of  grocery  clerk,  96 

Proceeds,   net,  255 

Product,  58 

Products    of    means    and    of    extremes, 
197 

Products    of    sums    and    differences    of 
lines,  297-299 

Products,    1897,    1902,    values   of   manu- 
factured and  industrial,  54 

Promissory  note,  265.  266 

Proper  fraction.  161 

Property,  personal,  342 

Property,  real,  342 

Proportion,   142-146,    196-200.      See   also 
similar  triangles 

Proportion    and    weight,    areas    and    vol- 
umes of  irregular  forms  by,  139,  140 

Proportion,     practical     applications     of, 
197-200 

Proportion,  terms  of.  196 

Protractor   (and  figure),  290-293 

Provisions,  prices  of.  8 

Pull.     Sec  tractive  force 

Pulse,   rapidity  of,  63 

Pure  decimal,  203 

Pyramid    (and  figure),   volume  of  trian- 
gular. 320 

Pyramid,  dimensions  of  Great,  320 

Pyramid,   model  and" development  of  tri- 
angular,  320 


Quadrant,  236,  289.  293 

Quadrilateral    (and   figure),   115,  "287 

Quadrilateral,  sum  of  angles  of,  292,  296 

Quadrillion,  32  and  foot-note 

Quart,  228 

Quart  of  milk,  weight  of  one,  22 

Quarter-round,  or  ovolo,  molding,  325 

Quintal.  239 

Quintillion,  32  and  foot-note 

Quire,   229 

Quotations   and    transactions,   stock  and 

bond,   345 
Quotient,  74 


Radical   sign,   328,   333 

Radius,  104 

Radius  of  earth,  mean.  33.  320 

Rail,    weight    of   one   yard    of    steel,    66, 

76.  79 
Railroad     cars,     engines      and      tenders, 

heights,     lengths,     and     weights    of, 

20,  76,  135,  136,  271 
Railroad   cars,   tractive  force  for  street, 

79.  80 

Railrcad   data,    street.    77 
Railroad  mileage  of  United  States.  1900, 

320 

Railroad  trains,   problems  on,   134-136 
Railroad   trains,  tractive  force  for,   135, 

136,  271-275 

Railroads,    mileage   and    number    of   em- 
ployees of  street,  77 


Rain  and  snowfall.  See  also  precipita- 
tion, 203,  204 

Rain  gauge,  figure  of,  GS 

Rainfall.  See  also  precipitation  and 
weather.  68-71 

Rainfall,   inch,  69,  203 

Rains,  numbers  of,  39.  See  also  precipi- 
tation. 

Range  (in  land  surveys),  314 

Range  of  barometer,   28 

Range  of  temperature,  23 

Rate,  242,   258 

Rate  of  running,  average,  138 

Rate  per  cent,  258 

Ratio.  141,  142.  See  also  similar  tri- 
angles, 142 

Ratio  and  as  equal  parts,  fraction  as, 
143-146 

Ratio  of  circumference  to  diameter,  214, 
215 

Rays,   study  of  sun's,   299-305 

Reading  numbers,  33,  34 

Reading  of  barometer,  26 

Reading  of  thermometer,  367 

Readings  of  thermometer,  positive  and 
negative,  369.  371 


Real  property,  342 
Ream.  229 


Reaumur  thermometer,   367-373 

Reciprocal,    180 

Rectangle  (and  figure),  115,  286-288 

Rectangle,  area  of,  45,  46,  70,  114,  116, 
189 

Rectangle,  properties  of,  297 

Rectangular  parallelepiped  (and  figure), 
volume  of,  70,  118,  119.  316 

Reduction,  sec  aiso  equivalents 

Reduction,  ascending,  230.  232 

Reduction,  descending,  230,  232 

Reduction  of  common  fraction  to  deci- 
mal, 218,  219 

Reduction  of  decimal  to  common  frac- 
tion, 202,  203 

Reduction  of  fractions,  146,  148,  151, 
154,  155,  161-163,  218,  219 

Reduction  of  fractions  to  higher,  lower, 
and  lowest  terms,  146-148 

Relation  signs,   354 

Remainder,   47 

Rent,   cash,  14 

Rent,  grain,  14 

Repetend.  219 

Rhomboid  (and  figure),  115,  189.  286, 
287 

Rhombus  (and  figure),  115,  286.  287 

Rifle  ball,  velocity  of,  216 

Right  angle  (and  figure),  282.  293 

Right  circular  cone,  volume  of,  318 

Right  circular  cylinder,  lateral  surface 
of,  317 

Right  circular  cylinder,  model  and  de- 
velopment of.  317 

Right    circular    cylinder,    volume   of,    317 

Right  prism,  model  and  development  of, 
280 

Right  prism   (square  pile),  figure  of,  281 

Right  prism,  volume  of,  281,  282 

Right  section,  221 

Right  triangle   (and  figure),  189 


402 


RATIONAL   GRAMMAR    SCHOOL   ARITHMETIC 


Right  triangle,  relation  of  sidess  of,  326, 
327,  331 

River   basins,    areas   of,    253 

Rivers,   lengths  of.   253 

Rivers,   rapidity   of,   216 

Road  wagon,  tractive  force  for,  79,  200- 
212,  274,  275,  352 

Rod    (unit)  ;    also   square   rod,    226,    227 

Roman  notation,  34,  35 

Roman   numeral,   34,   35 

Roof,  figure  of,  71,  312,  313 

Roof,  figure  of  house  and,  71,  313 

Roof,  model  and  development  of,  311 

Roofing  and  brick  work,  311-313 

Roofing,    square   of,    311 

Room,    model    and    development    of,    115 

Root,   short  method  for  square,  878 

Root,  square,  328 

Roots,  cubes  and  cube,  332,  333 

Roots  of  numbers  and  products  geomet- 
rically, 332 

Roots,   squares  and  square,  327-332 

Ropes,  data  concerning,  364 

Ruble,   225,   226 

Rules    (and  figure),  parallel,   187,  188 

Rye,  area  and  yield  of  Kentucky,  82 

Sail  areas  of  yachts  (and  figure),  224 

Saturn,  diameter  of,  321 

Scale   drawings,    1,   2,   5,   7,   11,    38,   91, 

114.    139,    140,    193-195.      See    also 

models   and   plotting   curves 
Scale  drawings  of  familiar  objects,  193- 

196 

Scalene  triangle   (and  figure),  109 
Scales,  figure  of,  103 
Schedule,   train,   137 
School  data,   Cuban,   40 
School  data  for  largest  citieg  of  United 

States,   55 
School   data   for   states,    territories,   etc., 

44 

School  grounds,  figure  of.  195 
School  house  and  grounds,  figure  of,  195, 

196 

School   room,   figure  of,  1 
Score,  229 
Scotia  molding,   325 
Scruple,   226 

Second  of  angle,  291,   298 
Second  of  arc,  291,  293 
Second  of  time,   229 
Section   of  land    (and  figure),    125,    227, 

315 
Section   of  land,    figure  of   divided,   125. 

315. 

Section,   right,  221 
Sextant,  260 
Shape,  likeness  of,  1,43 
Share  of  stock,  345 
Shears,  figure  of,  91 
Shilling,  225 
Ships.     See  battleships 
Skooting  stars,  25 
Short  division,   75,  76 
Short    methods   for   addition,   374 
Short  methods  for   division,   82-86,   201, 

377 
Short  methods  for  multiplication,  63-65, 

202.  375-377 


Short  methods  for  square  root,  378 
Short  methods  for  subtraction.  375 
Shortening    and     checking     calculations, 

methods  for,  373-378 
Sides  of  amgle,   292 

Sides   of  equation,   first  and  second,  103 
Sides  of  similar  triangles,  corresponding, 

oo4 

Sidewalk  assessment,  92 

Sieve,  figure  of  sand,  120 

Signs,  relation,  354 

Signs  : 

'  foot,  feet ;  minute  (s)  of  angle  and  of 

arc,   5,   293 
"  Inch,  inches  ;  second  (s)   of  angle  and 

of  arc,  5,  293     „ 
4-  plus,  36 

=  equal,  equals,  36,  108 
—  minus,  47 

X  times,  multiplied  by,  58 
-f-/ —  divided  by,  74,  7§,  154 
%  hundredth,  hundredths,  130,  242 
<,  >  is  less  than  ;   is  greater  than,   K2, 

35S 

iperpendicular  to,  325 
V  s_quare  root  of.  328 
iK      cube  root  of,  333 
Silver    coins,    fineness   of    United    States 

gold    and,    value    of   one   pound    of, 

67,  233 
Similar    triangles.      See    also    ratio    and 

proportion,    143,    333-340 
Similar  triangles,  uses  of,  198-200,  335- 

340 

Simple  decimal,   203 
Simple  interest,  131,  132 
Six  per  cent  method.  131.  132.  258 
Skiameter  (and  figure),  299-305 
Slant  of  sun's  rays,  changes  in.  300-305 
Sliding  and   static  friction,   364-367 
Sliding   friction,  law  of.   367 
Solar  day,  mean,   124.  229 
Solar  year,  mean,  229 
Solidus,  75 

Sound,  velocity  of.  67,  89,  216 
Sparrow,  speed  of,  216 
Specific   gravity,   217,   218 
Sphere,  area  of  surface  of,  319 
Sphere,  volume  of,  321 
Spirometer   (and  figure),  16,  17 
Sprocket  wheel.  222,  223 
Square  (and  figure),  115.  287,  327 
Square,  area  of,  114,  189 
Square  cornered  box,  volume  of,  69,  70, 

118 

Square,  inscribed,   192 
Square  of  number,   327 
Square   of  roofing,   311 
Square  prism,  model  and  development  of. 

279 
Square  prism,  volume  of,  118,  119,  281, 

282,  316 

Square  root,   328 
Square  root  by  subtraction.  378 
Square  root,  checking,  378 
Square  root  of  numbers  and  of  products 

geometrically,  332 
Square  root,  short  method  for,  378 
Square  roots,  squares  and,  327-332 
Square  unit,  45,  114 


GENERAL   INDEX 


403 


Squares  and  square  roots,  327-832 

Squares  of  units,  tens,  and  hundreds, 
table  of.  328 

Standard  base  line,  314 

Standard   time,   309-311 

Star,   distance   from   sun  to   nearest,    rf4 

Statement  of  the  problem.  58 

Statements  in  words  and  in  symbols, 
358-364 

States,  territories,  etc.,   areas  of,  44,  4 

States,  territories,  etc.,  dates  of  admis- 
sion and  populations  (1890)  of.  oo 

States,  territories.  etc.,  populations 
(1890,  1900)  of,  44,  53 

States,  territories,  etc.,   school  data  tor, 

Static  friction,  sliding  and,  364-867 
Stature    of    persons.      See    also    heights, 

etc.,  206,  207 
Statute  mile,  234 
Steamboat,  speed  of,  216 
Steel    rail,    weight   of   one   yard   of,    bb, 

Steel,  weight  of  one  cubic  inch  of,  119 
Stere,  239 

Stock'  and  bond  quotations  and  transac- 
tions, 345 

Stock  certificate,  345 
Stock,  common,  346 
Stock,   preferred,  346 
Stock,  sbare  of,  345 
Stocks  and  bonds,  345-347 
Stone,  measuring,  215,  227 
Straight  angle   (and  figure),  289 
Straight  line  law,  366 
Street   cleaning,   effect  upon   death   rate, 

19 

Street  crossing,  figure  of,  91 
Street   railroad   cars,   tractive   force  for 

79,  80 

Street  railroad   data,   77 
Street  railroad  mileage  and  numbers  ot 

employees,  77 

Streets  and  blocks,  figure  of  city,  116 
Study  of  sun's  rays,  299-365 
Substituting,  272 
Subtraction,  checking,  375 
Subtraction  compared,   division  and,    i 

73 

Subtraction  of  angles,  294 
Subtraction  (of  common  numbers),  46-o 
Subtraction  of  decimals.     Sec  also  deci 

mals,   205,  206 

Subtraction  of  fractions.      See  fraction 
Subtraction    of    fractions    with    commoi 

fractional  unit,  154 
Subtraction  of  literal  numbers,  56 
Subtraction,  short  methods  for,  375 
Subtrahend,  47 
Successive  division,  greatest  common  di 

visor  by,  151 

Sum' and  difference  of  angles.  296-29 
Sum   of   acute   angles  of   right  .jtnangle 

296 

Sum  of  angles  of  hexagon.  296 
Sum  of  angles  of  n-gon,  297 
Sum  of  angles  of  octagon,  297 
Sum   of   angles   of   parallelogram,   296 


um  of  angles  of  quadrilateral,  292,  296 
um  of  angles  of  triangle,  292-295 
urns   and  differences  of   lines,   products 
of,  297-299 

«Hm,  diameter  of,  321 

mn,  mass  of,  34 

>un,  mean  distance  from  earth  to,  dd 

Sun,  mean  radius  of,  33 
un  noon,  finding  time  of,  304 
un  time,  local,  306 

Sun's    light    and    heat,    distribution    of, 
299-805 

Sun's  rays,  changes  in  slant  of,  300-305 

^'s  rays,  study  of,  299-305 

plemental  angles,  295 
urface  area,  114-118.  227.  238 

Surface,  measuring.  114-118.  227,  238 

Surveying  land.  314-316,  335-340 

Surveyors'   measure,  227 

Switzerland,  population  and  area  of,  81 

Symbol.     See  sign 

Symbols,    statements    in    words    and    in, 
358-364 


Taxes.     See  also  assessment,  342,  343 
Tea  into  United  States,   imports  of,   41, 

r»o       oi  O 

Telegra'ph  wire,  cost  and  weight  of,  59, 

67 
Temperature.     See  also  thermometer  end 

weather,  22-25,  250 
Temperature,  measuring,  122,  123 
Temperature   of   a   man's    body,    normal, 

372 

Temperature,  range  of,  23 
Temperatures,  boiling,  122,  368,  372 
Temperatures,  freezing,  122,  368 
Temperatures,  mean,  368 
Temperatures,  melting,   368,  871 
Temperatures  of  gases,  critical,  372 
Ten-thousandth,  201 
Tenant  farmer,  14 
Tenders,    weights,    lengths,    and    weights 

of  railroad  cars  and  engines,  20,  7b, 

135,   136,  271 
Tenders,    water    and    coal    capacity    of 

locomotive.  271 
Tenth   (decimal),  201 
Terminal  growth  of  trees.  156,  Iu7 
Terms    of    fraction.      See     also     lowest 

terms,  145 

Terms  of  proportion,  196 
Territorial  growth  of  United  States,  252 
Territory  of  United  States,  area  of  ac- 
quired. 252 

Tests  of  divisibility  of  numbers,   80,   86 
Thermometer   -(and    figure),    centigrade. 

See  also  weather,  367-373 
Thermometer    (and    figure),    Fahrenheit, 

22-25.   122,  367-373 
Thermometer     (and     figure),     Reaumur, 

Thermometer,  graphically,  laws  of,  372, 
^73 

Thermometer,  positive  and  negative  read- 
ings of,  369,  371 

Thermometer,  range  of,  23 

Thermometer,  reading  of,   367 


404 


RATIONAL    GRAMMAR    SCHOOL    ARITHMETIC 


Thermometer.     See  also  weather,   22-25, 

367-373 
Thermometers,     equivalent     readings    of, 

371-373 

Thermometers,  graduation  of,   367-370 
Thousand,  32 
Thousandth,  201 

Time  belts  of  United  States,  map  of,  301) 
Time,  central,  309 
Time,  eastern,  309 

Time,  equivalents  of  longitude  and,   306 
Time,  local   (sun),  306 
Time,  longitude  and.  306-311 
Time,  measuring,   124,   125,  229 
Time,  minute  of,  229 
Time,  mountain,  309 
Time,  Pacific,  309 
Time,   second  of,  229 
Time,   standard,   309-311 
Time,  sun,   306 
Time  to  run    (phrase),  265 
To  the  weather    (phrase),  311 
Ton,   226 
Ton,   long.   226 
Ton,  metric,  239 
Ton,  number  of  cubic  feet  of  hay  and  of 

coal  in,  215 
Torus   molding,   325 
Tower,  model  and  development  of,  313 
Town   block   and   lots    (and   figure),   2-4, 

91,    92 
Township    (and   figure).    126,    227,    314, 

315 
Tract  of  land,  figure  of,  11.  38,  91,  116. 

125,    130,    195,    198,    199,    314,    315, 

336-339 
Tractive   force  for   railroad  trains,   135, 

136,    271-275 
Tractive  force  for  road  wagon,   79,  209- 

212,  274,  275,  352 
Tractive   force   for   street   railroad    cars, 

79,  80 

Trade  discount,  255,  256.  343,   344 
Trade,    1891,    1901,    United    States    ex- 
port,  54 

Trailers  of  locomotive,  270 
Train  despatched  report,  136-138 
Train  schedule,  137 

Trains  at  Chicago,   number  of  daily,   75 
Trains,    freight    and    passenger,    134-136 
Trains,  problems  on   railroad,   134-136 
Trains,   speed  of  express,   88,  89 
Trains,  tractive  force  for  railroad,   135, 

136,  271-275 
Transportation  problems,  applications  to. 

See  also  tractive  force.  270-275 
Trapezium   (and  figure),  115 
Trapezoid,  altitude  of,  117 
Trapezoid   (and  figure),  115,  287 

Area  of,  117,  118 

Bases  of,  117 

Dimensions  of,  118 
Trees,  growth  of,  155-157 
Trees,  lateral  growth  of,  156,  157 

Lateral  growth  of.  156,  157 

Leaves  of,   205,   249 

Terminal  growth   of,  156,   157 
Triangle,  altitude  of.  117 
Triangle    (and   figure),    equilateral,    108, 

109 


Triangle  (and  figure),  isosceles,  108,  109 

Triangle   (and  figure),  right,  189 

Triangle   (and  figure),  scalene,   109 

Triangle,   area  of,   117,  189 

Triangle,  base  of  isosceles,  108 

Triangle,  relation  of  sides  of  right,  326, 
327,  331 

Triangle,  sum  of  angles  of,  292,  295 

Triangles  (and  figures)  and  their  uses, 
similar,  143.  198-200,  333-340 

Triangles  (and  figures),  uses  of  the  30° 
and  60°,  and  of  the  45°  right,  190- 
192 

Triangular  prism  (and  figure),  volume 
of,  320 

Triangular  prism,  model  and  develop- 
ment of,  281 

Triangular  pyramid  (and  figure),  volume 
of,  320 

Triangular  pyramid,  model  and  develop- 
ment of,  320 

Trillion,  32  and  footnote 

Trisect,  188 

Troy  weight,  226 

True  discount,  266 


Unit,  32 

Unit,  concrete,  225 

Unit,  fractional,  133,  144 

Unit,  square,  45,  114 

United  States,  land  areas  of  divisions  of, 

211 

United  States,  maps  of.  43,  809 
United  States  notation,  32  and  note 
United  States  Rule  for  Partial  Payments, 

268 

United  States,  territorial  t growth  of,  252 
United-  States,   1790-1900,  population  of, 

129 

United  States.  1900.  distribution  of  pop- 
ulation of,  43-45 
Uranus,  diameter  of,  321 
Use     of    letters    to    represent    numbers, 

349-364 

Uses  of  numbers,  common,  133-140 
Uses  of  similar  triangles,   198-200,   335- 

340 

Uses  of  the  equation.  352.  364-373 
Uses  of  the  30°   and  60°.   and   the  45°, 

right  triangles,   190-192 


Valuation,  assessed,  342 

Value,  face,  265 

Value,  measuring,  225,  226 

Value  of  digit,  name,  31 

Value  of  digit,  place,  31 

Value  of  x,  58 

Values  of  cattle,  weights  and,  68 

Venus,  diameter  of,  321 

Vertex,  186,  278.  292 

Vertex  of  angle,  186,  292 

Vertical  angles,  opposite  or,  292 

Vertices.     Plural  of  vertex 

Vibration  for  colors,   rates  of,   34 

Vital   statistics,   19 

Volume,  316 

Volume,    measuring,    118.    119,    227,    238 

Volumes,   316-321 


GENERAL    INDEX 


405 


Wagon,  tractive  force  for  road,  79,  209- 

212,  274,  275,  352 
Wall,  figure  of  brick,  313 
Walls  of  house,  4-6 
Watch,  figure  of,  124,  289 
Water  area  of  earth,  252 
Water,  weight  of  one  cubic  foot  of,  175, 

271 

Water,  weight  of  one  cubic  inch  of,  174 
Weather,     tiec  also  barometer,  heat,  light, 

meteorology,    precipitation,    rainfall, 

temperature,  thermometer,  wind,  250 
Weather   Bureau  stations,  elevations  of, 

249 

Weather,  to  the   (phrase),  311 
Week,  229 

Weight,  apothecaries',  226 
Weight,   areas   and  volumes  of  irregular 

forms   by   proportion  and,   139,   140 
Weight,   avoirdupois,   226 
Weight,    growth    of    boys    and    girls    in 

height  and,  18,  127'.  128.  206,  207 
Weight,    measuring,    120-122,    226,    238. 

239 
Weight    of    one    cubic    foot    of    various 

things,  88,  119.  217.  218 
Weight  of  one  cubic  inch  of  steel,  119 
Weight,  troy,  226 


Well  lighted   (room),  7,  114 

Wheat  yield  and  area  of  Dakotas,  82 

Wheel,  sprocket,  222,  223 

Wild  duck,  speed  of,  216 

Wind  movement.     See  also  weather,  248, 

249 

Wind  pressure,   19,  20 
Wind,  velocity  of,  19,  20,  248 
Winds,  names  of,  19 
Wire,  cost  and  weight  of  fence,  11,  12 
Wire,   cost  and  weight  of  telegraph,   59, 

67 

Wood  (not  fire-wood),  measuring,  12 
Worth,  present,  266 
Writing  numbers,  34 

Yachts,  sail  areas  of,  224 

Yard  ;   also  square  and  cubic  yard,  226, 

227 

Year,  229 

Year,  calendar,  229 
Year,  common.  229 
Year,  leap,  229 
Year,  mean  solar,  229 

Zenith,  304 
Zero,  31 


YC  49548 


VERSITY  OF  CALIFORNIA  LIBRARY 


